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1 NOTE CAREFULLY The following document was developed by Learning Materials Production, OTEN, DET. This material does not contain any 3 rd party copyright items. Consequently, you may use this material in any way you like providing you observe moral rights obligations regarding attributions to source and author. For example: This material was adapted from (Title of LMP material) produced by Learning Materials Production, OTEN.

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3 Mathematics Stage 4 CENTRE FOR LEARNING INNOVATION Gill Sans Bold MS4.1 Perimeter and area M7/

4 The Centre for Learning Innovation gratefully acknowledges the following owners of copyright material for permission to reproduce their work. All reasonable efforts have been made to obtain copyright permissions. All claims will be settled in good faith. Unless otherwise identified, all unacknowledged aspects of this material, including text and graphics, have been created by Centre for Learning Innovation, NSW Department of Education and Training. Writer Jim Stamell Editor Ric Morante Illustrator Thomas Brown, Tim Hutchinson Desktop publisher Gayle Reddy Version date April Revised June September 20, 2004 Produced by the Centre for Learning Innovation, 51 Wentworth Road, Strathfield NSW Reproduction or transmittal in whole, or in part, other than in accordance with provisions of the Copyright Act 1968 is prohibited without the written authority of the Centre for Learning Innovation. State of New South Wales, Department of Education and Training, 2004.

5 Unit contents Unit overview... iii Outcomes... iii Indicative time...vi Resources... vii Icons... viii Glossary...ix Part 1 Length and measurement Part 2 Perimeter and Pythagoras Part 3 Pythagoras s theorem Part 4 Areas Part 5 Further area and perimeter Part 6 About circles Unit evaluation...53 Unit overview i

6 ii MS4.1 Perimeter and area

7 Unit overview In this unit you will use formulas and Pythagoras s theorem in calculating the perimeter and area of circles and figures composed of rectangles and triangles. Some of the key ideas involved in this unit include: describing the limits of accuracy of measuring instruments developing formulae and using them to find the area and perimeter of triangles, rectangles and parallelograms finding the areas of simple composite figures applying Pythagoras s theorem investigating and finding the area and circumference of circles converting between metric units of length and area. Outcomes By completing the activities and exercises in this unit, you are working towards achieving the following outcomes. You have the opportunity to learn about: Length and perimeter estimating lengths and distances using visualisation strategies recognising that all measurements are approximate describing the limits of accuracy of measuring instruments ( ± 0.5 unit of measurement) interpreting the meaning of the prefixes milli, centi and kilo converting between metric units of length finding the perimeter of simple composite figures. Unit overview iii

8 Pythagoras s theorem identifying the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle establishing the relationship between the lengths of the sides of a right-angled triangle in practical ways, including the dissection of areas using Pythagoras s theorem to find the length of sides in right-angled triangles solving problems involving Pythagoras s theorem, giving an exact answer as a surd (eg 5 ) and approximating the answer using an approximation of the square root writing answers to a specified or sensible level of accuracy, using the approximately equals sign identifying a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square of the third using the converse of Pythagoras s theorem to establish whether a triangle has a right angle Areas of Squares, Rectangles, Triangles and Parallelograms developing and using formulae for the area of a square and rectangle developing (by forming a rectangle) and using the formula for the area of a triangle finding the areas of simple composite figures that may be dissected into rectangles and triangles developing the formula by practical means for finding the area of a parallelogram eg by forming a rectangle using cutting and folding techniques converting between metric units of area 1 cm 2 = 100 mm 2, 1 m 2 = mm 2, 1 ha = m 2, 1 km 2 = m 2 = 100 ha Circumferences and Areas of Circles demonstrating by practical means that the ratio of the circumference to the diameter of a circle is constant eg by measuring and comparing the diameter and circumference of cylinders iv MS4.1 Perimeter and area

9 defining the number ϖ as the ratio of the circumference to the diameter of any circle developing, from the definition of ϖ, formulae to calculate the circumference of circles in terms of the radius r or diameter d C = π d or C = 2π r developing by dissection and using the formula to calculate the area of circles. You have the opportunity to learn to: consider the degree of accuracy needed when making measurements in practical situations (Applying Strategies) choose appropriate units of measurement based on the required degree of accuracy (Applying Strategies) make reasonable estimates for length and area and check by measuring (Applying Strategies) select and use appropriate devices to measure lengths and distances (Applying Strategies) discuss why measurements are never exact (Communicating, Reasoning) describe the relationship between the sides of a right-angled triangle (Communicating) use Pythagoras s theorem to solve practical problems involving right-angled triangles (Applying Strategies) apply Pythagoras s theorem to solve problems involving perimeter and area (Applying Strategies) identify the perpendicular height of triangles and parallelograms in different orientations (Communicating) find the dimensions of a square given its perimeter, and of a rectangle given its perimeter and one side length (Applying Strategies) solve problems relating to perimeter, area and circumference (Applying Strategies) Unit overview v

10 compare rectangles with the same area and ask questions related to their perimeter such as whether they have the same perimeter (Questioning, Applying Strategies, Reasoning) compare various shapes with the same perimeter and ask questions related to their area such as whether they have the same area (Questioning) explain the relationship that multiplying, dividing, squaring and factoring have with the areas of squares and rectangles with integer side lengths (Reflecting) use mental strategies to estimate the circumference of circles, using an approximate value of ϖ (Applying Strategies) find the area and perimeter of quadrants and semi-circles (Applying Strategies) find radii of circles given their circumference or area (Applying Strategies) solve problems involving ϖ, giving an exact answer in terms of ϖ or an approximate answer (Applying Strategies) compare the perimeter of a regular hexagon inscribed in a circle with a circle s circumference to demonstrate that ϖ > 3 (Reasoning) Source: Extracts from outcomes of the Maths Years 7 10 syllabus < 710_syllabus.pdf > (accessed 04 November 2003). Board of Studies NSW, Indicative time This unit has been written to take approximately 24 hours. Each part should take approximately 4 hours. Your teacher may suggest a different way to organise your time as you move through the unit. vi MS4.1 Perimeter and area

11 Resources Resources used in this unit are: For part 1 you will need: a calculator a ruler For part 2 you will need: a calculator a ruler a protractor access to the Internet For part 3 you will need: a calculator a ruler For part 4 you will need: a calculator a ruler For part 5 you will need: a calculator a ruler access to the Internet For part 6 you will need: a calculator a ruler Unit overview vii

12 Icons Here is an explanation of the icons used in this unit. Write a response or responses as part of an activity. An answer is provided so that you can check your progress. Compare your response for an activity with the one in the suggested answers section. Complete an exercise in the exercises section that will be returned to your teacher. Think about a question or problem then work through the answer or solution provided. Access the Internet to complete a task or to look at suggested websites. If you do not have access to the Internet, contact your teacher for advice. Perform a practical task or investigation. viii MS4.1 Perimeter and area

13 Glossary The following words, listed here with their meanings, are found in the learning material in this unit. They appear in bold the first time they occur in the learning material. For these words and their meanings including pronunciation see the online glossary on the LMP website at and follow the links to Stage 4 mathematics. apex area circumference composite shape converting units Diameter estimate formula fractal hectare hypotenuse irregular shape parallelogram perimeter perpendicular plane figure Pythagoras s theorem The highest point relative to some line called the base. The amount of surface. It is usually measured in square units, such as square centimetres, square millimetres, square metres, and square kilometres. The name given to the perimeter of a circle. A shape made up of several simple shapes. Changing measurements from one set of units to another. The distance across a circle through its centre. A reasonable guess based on knowledge and experience A rule expressed in algebraic symbols. Any pattern that reveals greater complexity as it is enlarged. A metric unit of surface area (symbol ha), equal to m 2. It is used for measuring larger areas such as land. The longest side of a right-angled triangle, which is the side opposite the right angle. A plane shape that is not made up of simple shapes. A quadrilateral where the opposite sides are equal in length and parallel to each other. The perimeter of a plane figure is the sum of the lengths of the sides. A line drawn at right angles to another line. Any flat shape. The rule which states that in any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Unit overview ix

14 Pythagorean triad quadrants radius rhombus semicircle surd vertex A set of positive integers (a, b, c) where a 2 + b 2 = c 2. For example, (3, 4, 5) form a Pythagorean triad since = 5 2. A quarter of a circle. The distance from the centre of a circle to the rim. A special kind of parallelogram where all four sides are equal in length. Half of a circle. A circle divided into two parts by cutting along a diameter. A number that cannot be expressed completely as a decimal. It is most accurately expressed using a root sign. The corner of a plane (flat) shape. x MS4.1 Perimeter and area

15 Centre for Learning Innovation NSW Department of Education and Training