Students Integrated Maths Module for Indirect Measure 1

Transcription

1 Students Integrated Maths Module for Indirect Measure 1 Author: Steve Hadfield Editor: Mark R. O Brien

2 OTRNet Publications Check our website for contact details. Copyright 2002 by OTRNet Publications, All rights reserved. No part of this publication may be reproduced or transmitted in any form, or by any means, electronic or manual, including photocopying, scanning, recording, or by any information storage or retrieval system, without permission in writing from the publisher. First published: October 2002 First reprinted: February 2005 Revised: January 2019 Design and Editing: Mark R. O Brien Cover Design: Ali B Design Ph: ali@alib.com.au National Library of Australia Cataloguing-in-Publication data For secondary students ISBN

3 3 Introduction for students: At the end of this module you should have increased your ability to; Make sensible estimates of unknown measurements using given information Use scale drawings to find measurements approximately Apply the properties of similarity to find unknown measurements Apply the Pythagorean theorem to find unknown measurements Apply the tangent ratio to find unknown measurements Table of Contents: Activities: Exercises: (cont.) A: Direct Distance 4 6: Solving Right Triangles 31 B: Pythagorean Cut Out 5 7: Using the Tangent Ratio 32 C: Phone Towers 8 D: Duck Island 12 Puzzles: E: Grids and Dilations 14 In The Army: Part A 34 F: Image Projections 16 In The Army: Part B 35 G: Grid Distance 18 H: Investigation: Pythagorean Triads 20 Applications: A: Positioning The Troops 36 Student Recording 21 B: In The Himalayas 37 C: Train Tracks 40 Notes 22 D: Surveying 42 Exercises: 1: Scales and Conversions 24 E: Project: Passive Solar Design 43 F: Newspaper Study: Getting An Angle On It 45 2: Maps and Scales 25 Student Reflection 47 3: Right Triangle Calculations 27 4: Right Triangles and Problems 28 Answers to Exercises 48 5: The Tangent Ratio 30

4 4 Activity A: Materials required: Ruler, protractor, blank A4 paper Direct Distance The sketch of Queensland here shows three towns in the North of the state. The sketch shows that Normanton is 332 km North of Cloncurry and that Hughenden is 420km East of Cloncurry. If a road was built directly from Normanton to Hughenden how much distance would be saved travelling between the two? Task 1: Using the distances shown here, how far is it from Normanton to Hughenden via Cloncurry? Task 2: Task 3: Task 4: Estimate the direct distance from Normanton to Hughenden. Draw a large, scale diagram showing the three towns. Using your scale diagram find the direct distance from Normanton to Hughenden. The next series of tasks will introduce you to a method for finding this distance exactly, using calculations only. This method was developed about 2500 years ago by a mathematician of the time known as Pythagoras and is known as the Pythagorean Theorem. Task 5: Find the following values, and Task 6: Add together these two values ( ). Task 7: Find the square root of the sum ( ). Task 8: Task 9: How accurate was the value you found from your scale drawing? How much distance would be saved with a direct road?

5 5 Activity B: Materials required : Scissors, ruler, protractor, blank A4 paper. Pythagorean Cut Out Task 1: Draw an accurate copy of the diagram drawn here. It consists of a right angled triangle with squares on each side. Task 2: Calculate the area of each of the three squares. Task 3: Cut out the 4cm by 4cm square and place it on the 5cm by 5cm square. Task 4: Now cut out the 3cm by 3cm square and by cutting it into pieces place it onto the uncovered part of the 5cm by 5cm square. Task 5: Draw a diagram to show how this was done including measurements. Task 6: Explain how the areas calculated in task 2 demonstrate that task 4 must be possible.

6 6 Task 7: Using calculations, or by cutting out a diagram, show that this would not be possible with a triangle of sides 4cm, 5cm and 6cm. Task 8: Another right triangle has sides of 6, 8 and 10 units as shown. What would be the areas of squares drawn on each side? Task 9: Would the sum of the areas of the squares on the shorter sides be equal to the area of the square on the longer side for this triangle? Task 10: What feature do the triangles in tasks 1 and 8 have that the triangle in task 7 does not have? Task 11: Complete this table to record the information from the two right triangles in tasks 1 and 8 and the three on the top of the next page. Use h as the hypotenuse (the longest side, opposite the right angle) and a and b as the other two sides. Triangle Sides Areas of Squares on Sides a b h a 2 b 2 h 2 a 2 +b

7 7 Task 12: Refer to the last two columns of the table. Comment on the values in these columns. Task 13: Based on your work above, which one of these two triangles would you expect to be right angled? Explain your choice. (Note: The diagrams are not drawn to scale.)

8 8 Activity C: Materials required : Ruler, protractor, blank A4 paper Phone Towers In an attempt to find the height a phone tower measurements were taken using a tape measure and inclinometer. This sketch shows the measurements recorded for the tower. The angle of inclination was 42 at a distance of 30m from the centre of the base of the tower. Task 1: Using these values make an estimate of the height of the tower. Explain how you made this estimate. Task 2: Draw an accurate scale diagram of the sketch above and use it to find the height of the tower. It is possible to use the fact that all right triangles with the same angle sizes have the same side ratios to find the height of the tower exactly without any drawing. The following tasks will use this fact. Task 3: Using your calculator find the value of the tangent of 42. This involves using the key usually labelled as TAN. Write this value down to four decimal place accuracy.

9 9 Task 4: Now multiply this value by 30m. Task 5: Next add on the eye height of the person using the inclinometer to get the tower height. The following tasks will develop your understanding of the tangent ratio and its use. Task 6: On a blank sheet of paper accurately draw four large right triangles with an angle of 14. Make each one a different size. Task 7: Clearly mark the right angle and the angle of 14 in these triangles. Task 8: Label the sides of the triangles according to these rules: Label the hypotenuse of each triangle HYP. This is the side opposite the right angle, it will be the longest side. Label the side adjacent to the 14 angle ADJ. This is the side next to the angle but it is not the hypotenuse. Label the side opposite the 14 angle OPP. This side is furthest from the angle.

10 10 Task 9: Accurately measure the lengths of the sides opposite and adjacent to the 14 angle in each triangle and record them in this table. Triangle Angle OPP ADJ A B C D Task 10: Complete the last column of the table using your calculator. Task 11: Comment on the values in the last column. Task 12: What is the mean of the four values in the last column of your table? Task 13: Use your calculator to find the value of the tangent of 14 accurate to four decimal places. Task 14: Compare the mean value of your four calculations with the value from the calculator. How accurate were your measurements? Task 15: The other angle in each of the triangles you have drawn is 76. Why? In the next part of this activity we will look at the tangent ratio for the 76 angle in each of the four triangles. Task 16: Re-label the opposite (OPP) and adjacent (ADJ) sides of each of your four triangles so that they are opposite and adjacent to the 76 angle.

11 11 Task 17: Accurately measure the lengths of the sides opposite and adjacent to the 76 angle in each triangle and record them in this table. Triangle Angle OPP ADJ A B C D Task 18: Complete the last column of the table using your calculator. Task 19: Comment on the values in the last column. Task 20: What is the mean of the four values in the last column of your table? Task 21: Use your calculator to find the value of the tangent of 76 accurate to four decimal places. Task 22: Compare the mean value of your four calculations with the value from the calculator. How accurate were your measurements? Task 23: Look at this sketch of another right triangle. Explain how you could find the value for OPP ADJ in this triangle for the 50 angle without any measuring? Task 24: What is the value of OPP ADJ for 50?

12 12 Activity D: Materials required : ruler Duck Island This map shows some of the main features of Duck Island in a scale drawing. Task 1: Measure the direct map distance between the towns of Muscovy and Teal. It is known that the actual distance between the towns is 6 kilometres. Task 2: Complete the following for the above map. 3 cm : km 1 cm : km 1 cm : m 1 cm : cm Map scale is 1 :

13 13 Task 3: Use this scale to find the actual direct distance between the towns of Muscovy and Mallard. Task 4: Write the following scales without units in the form 1: X. 4 cm : 800 m 5 cm : 5 km 2 cm : 10 km Task 5: Measure each of these objects and use the actual measurement to form a scale for each. Write your scales in the form 1: X. Height 22.5 metres Length 4.2 metres Heights: Man 1.85m, Woman 1.75m Length 15 metres

14 14 Activity E: Materials required: Grid paper Grids and Dilations The three elephants shown above are all of the same picture but they are different sizes. They are said to be similar in shape. One method of dilating (enlarging or reducing) a simple drawing is to use a grid of squares to copy the original to the required size. Task 1: Enlarge and reduce this drawing of a shark to the given grids. Task 2: What scales have been used in the dilations of this shark?

15 15 Task 3: Use the grid here to dilate this clown. Task 4: What scale is being used for this dilation? Task 5: Use grid paper to produce dilations of these drawings below or those of your own choice. Use a variety of scales for your dilation grids.

16 16 Activity F: Materials required : ruler Image Projections The diagram below shows two similar triangles. Task 1: Task 2: Task 3: Draw lines through corresponding vertices (A & D, B & E, C & F) of the triangles and extend these to each side of the page. Label the intersection point of the three lines P. This is known as the projection point. Measure the lengths of the sides of each triangle. Task 4: Find the ratio of the corresponding side lengths of the larger triangle to the smaller triangle. This is the scale factor of the dilation. Task 5: Task 6: Measure the following lengths. PA = PB = PC = PD = PE = PF = How do the distances from P to each vertex of the larger triangle compare to those of the smaller triangle?

17 17 Task 7: Task 8: One of the projection lines has been drawn in for this figure. Draw in the other three. Use measurement along these lines to produce three dilations of the original figure with scale factors 2, 3 and. Task 9: What is the effect of a scale factor of 1? Task 10: Use three different projection points, including one inside the figure, for the triangle below to show that a scale factor of 2 will always produce the same sized result irrespective of the position of the projection point.

18 18 Activity G: Grid Distance These diagrams show two points A (2,3) and B (4,7) on the Cartesian plane. The second diagram also shows horizontal and vertical distances from one point to the other. Task 1: Where did the values of 2 and 4 come from in the second diagram? Task 2: Why do these lines form part of a right triangle? Task 3: Using Pythagorean theorem, calculate the distance between the points A and B. Task 4: This diagram shows the points C(3,2) and D(8,6). What are the horizontal and vertical distances between these points? Task 5: Calculate the distance between the points C and D.

19 19 Task 6: This diagram shows the points E(2,6) and F(7,3). What are the horizontal and vertical distances between these points? Task 7: Calculate the distance between the points E and F. Task 8: Draw a diagram to show the points (-4, 6) and (3, -4). Task 9: What are the horizontal and vertical distances between these points? Task 10: Calculate the distance between these two points. The final tasks require you to generalise this type of calculation. Using point A with coordinates (x 1, y 1 ) and point B with coordinates (x 2, y 2 ): Task 11: Using x 1 and x 2 write an expression for the horizontal distance between A and B. Task 12: Write an expression for the vertical distance between A and B. Task 13: Now using Pythagorean theorem write a formula for calculating the distance (d) between (x 1, y 1 ) and (x 2, y 2 ). Task 14: Show that your formula gives a distance of 7.62 units between the points (5, -3) and (8, 4). Task 15: Use the library or Internet to find out what Analytic Geometry is and describe it here.

20 20 Activity H: Materials required : A spreadsheet for generating triads is available from the TLR page of our web site at the following address: Investigation : Pythagorean Triads Pythagorean triads are the sets of three whole numbers that form right triangles. They can be generated by using the following formulae to give the lengths of the three sides. h = n (hypotenuse) a = 2n and b = n 2 1 where n is a whole number greater than one. Generate the Pythagorean Triads for n values of 0 to 24 and investigate these triads with the following points in mind. Why don t values of 0 or 1 generate useful results? Why is n 2 +1 always the length of the hypotenuse? When will 2n be the shortest side? Are any triads multiples of others? Are there groups of triads with the same ratio between the sides? Why is it that some triads such as 9, 40 and 41 are not able to be generated from the formulae? How are these obtained? When can a generated triad be simplified?

21 21 Student Recording: Write at least two pieces of information about each of these concepts that you have explored in earlier lessons. Then try to give an example relating to each. Use diagrams where it helps. hypotenuse: tangent: similar: scale: scale drawing/diagram: Pythagorean theorem: dilation:

22 22 Notes Scale Diagrams Objects which are too large or too small to be drawn to correct size can be represented using a scale diagram. Examples: Scale 3 : 1 Scale 1 : 500 Drawing is 3 times the size Windmill is 500 times the size of the insect. of the drawing. Scale diagrams produce similar figures which are the same shape but not the same size as the original. These are also known as dilations. Dilations can be made using grids or projections to produce larger or smaller versions of the original. Pythagorean Theorem Named after Pythagoras, a Greek mathematician from the 6th century BC. Pythagoras group of followers also developed the idea that the angles of a triangle sum to 180.

23 23 Right Triangles and the Tangent Ratio The tangent ratio can be used in right triangles to find unknown sides and angles. Tangent is one of the three trigonometric ratios used for right triangles, the other two are Sine and Cosine. The tangent of any given angle is the ratio of the opposite side to that of the adjacent side in relation to that angle. This abbreviates to the formula : tan è = (Memorise this ratio!) Note: When choosing variables for angles it is a convention to use variables from the Greek alphabet. Some common ones are: è - theta, á - alpha, â - beta. Examples : 1. Finding the opposite side. tan 26 = k = 16 x (tan 26 ) i.e. k = 7.80 m 2. Finding the adjacent side. tan 52 = b = i.e. b = 6.25 cm 3. Finding the angle. tan â = â = tan -1 ( ) i.e. â = (tan -1 is known as inverse tan or arctan - it is often shown as atan.)

24 24 Exercise 1: Scales and Conversions 1. Complete each of the following tables given the scale for each drawing. Use the nominated units for each conversion. (a) (b) Scale 1mm:1m Drawing 5mm 6cm 2.5mm mm cm cm 0.6cm Real m m m 8m 10m 200m m Scale 1cm:200m Map 2cm 30mm 8cm cm cm 1mm cm Actual m m km 500m 20km m 2.5km (c) Scale 1:5000 Map 4cm 8cm 45cm 5.5cm cm cm 0.6cm Actual m m km m 880m 2.8m m 2. Find the equivalent scale for these examples: Unit Scale Ratio Scale eg 1cm : 1m 1:100 (a) 1mm : 10m (b) 1mm : m 1 : 2000 (c) 1cm : 1km (d) 1cm : m 1 : 5000 (e) 1mm : 400m (f) 1mm : m 1 : (g) 1cm : 50km 3. What distance does one centimetre represent on each of these maps? (a) A city map has a scale of 1 : (b) A shire map with a scale of 1 : (c) An Australian map shows a scale of 1 : (d) A world map that has a scale of 1 :

25 25 Exercise 2: Maps and Scales This map shows a section of Eastern Europe and has a scale of 1 : Write this scale in unit form. 1 cm : km (Check this before continuing.) 2. Use the map to find the direct distances between these cities. (a) Riga and Tallinn (b) Mensk and Vilnius (c) Helsinki and Mensk (d) Kaliningrad and St. Petersburg 3. Estimate the dimensions (length and width) of the island of Gotland.

26 26 This map shows the region known as the Middle East and has a scale of 1 cm : 160 km. 4. Write this scale in ratio form i.e. 1: X. 5. Find approximate distances between these cities. (a) Cairo and Damascus (b) Baghdad and Kuwait (c) Amman and Riyadh (d) Adana and Jiddah 6. Find the approximate dimensions of the country of Lebanon. 7. Find the approximate length of the island of Crete.

27 27 Exercise 3: Right Triangle Calculations Find the length of the missing side for each of the right triangles below. Write your answers in decimal form accurate to two decimal places. Find the length of the sides marked with a variable accurate to two decimal places.

28 28 Exercise 4: Right Triangles and Problems 1. A rectangle has a length of 14cm and a width of 8cm. Find the length of the diagonals of this rectangle. 2. A square has a length of 11mm. Find the length of the diagonal of the square. 3. Find the width of a rectangle with a length of 20cm and a diagonal of 24cm. 4. Find the length of a square which has a diagonal of 16cm. 5. A hiker travels 7km North and then 5km East. Draw a sketch to illustrate this situation and then find the direct distance the hiker is from her starting point. 6. A 4 metre ladder leans against a wall and the foot of the ladder is 1.2 metres from the base of the wall. How far up the wall does the ladder reach?

29 29 7. A rectangular gate measuring 1.2 metres by 70 centimetres is sagging and requires a brace attached to a diagonal. What length of bracing is needed? 8. A man standing at a crossroad notices a truck 700 metres South of him and a car 450 metres West of him. What is the direct distance between the two vehicles? 9. A power cable runs between two buildings of heights 29 and 37 metres. If the buildings are 14 metres apart, find the length of the cable. 10. Pete the paver has completed this section of paving and wants to check to see if his work is square. He measures AB as 6.7 metres, BC as 3.4 metres and AC as 7.7 metres. Would he consider his paving as square? 11. A 40 metre radio mast is supported by two wires each 30 metres long. One is attached to a point 25 metres up the mast and the other 27 metres up the mast. How far from the base of the mast are the two wires anchored to the ground?

30 30 Exercise 5: The Tangent Ratio Use the tangent ratio to find the opposite side to the given angle in each triangle below. Find the adjacent side to the given angle in each of these triangles. Find the marked angle using the tangent ratio. Find the marked side or angle in each of these right triangles.

31 31 Exercise 6: Solving Right Triangles Solve each of these right triangles by finding all missing sides and angles. You will need to use both the tangent ratio and the Pythagorean theorem. 7. A right triangle has sides of 12, 16 and 20 centimetres. Find the size of each angle in the triangle. 8. A right isosceles triangle has a hypotenuse length of 12 metres. Find the lengths of the other sides.

32 32 Exercise 7: Using the Tangent Ratio 1. A ladder leans against a wall at an angle of 62 with the foot of the ladder 2 metres from the wall. How far up the wall does the ladder reach? 2. A support wire for a tree is attached to the ground 2.8 metres from the base of the tree and to the tree 2.4 metres up the trunk. Find the angle the wire makes with the ground. 3. From a ship out at sea the top of a lighthouse has an angle of elevation of 7. If the top of the lighthouse is 90 metres above sea level, how far out to sea is the ship? 4. A triangle is drawn in a semi circle creating a right angle at B. If the length of BC is 12 centimetres and angle BCA is 36, find the radius of the circle.

33 33 5. A hiker travels 5.5 kilometres North and then 6 kilometres East. What is the bearing of the hiker from his starting point? 6. A pyramid has a square base of 145 metres and the faces of the pyramid make an angle of 52 with the ground. Find the height of the pyramid. 7. A television mast has two wire supports on each side both at an angle of 64 to the ground. If the wires are attached to the mast at heights of 36 and 58 metres, find the distance of each on the ground from the base of the mast. Find also the length of each wire. 8. Find the height of an equilateral triangle of side 14 centimetres.

34 34 Puzzle: In The Army Part A : Why were the soldiers always tired at the start of April? Find the length of the missing side on each right triangle accurate to two decimal places. Match each of these in the code below to solve this puzzle

35 35 Part B : Why did the soldiers salute the refrigerator? Find the required angle or side using the tangent rule and round accurate to one decimal place. Match these in the code boxes to solve this puzzle

36 36 Application A: Positioning the Troops The robot warriors of Kyzar are great fighters but are hopeless when it comes to direction. They can only locate their battle positions by moving in the four directions North, East, South and West. Their controllers at Headquarters (HQ) however need to know the positions of the warriors in terms of bearings and distances. Task 1: Draw a scale diagram that shows the movement of Warrior á given that it moves 3km North and 7km West. Task 2: Use your scale diagram to measure the bearing and distance of Warrior á from HQ. Task 3: Calculate the exact bearing and distance of Warrior á from HQ. Task 4: Draw the positions of the following five warriors on a scale diagram then calculate the exact bearing and distance of each of these warriors from HQ. Check your answers against your diagram. Warrior â Warrior ä Warrior ç Warrior è Warrior ë 4km East and 8km North 7km South and 5km East 9km West and 4km South 5km West and 5km North 10km North and 3km West Task 5: In task 1, Warrior á was given a set of instructions to find its battle location. Give another set of instructions that would get it to the same position?

37 37 Application B: In The Himalayas The sketch below shows a view of the Annapurna Range of the Himalaya Mountains near Pokhara in Nepal. There are seven mountain peaks in this range and they are shown here along with their altitudes above sea level. Task 1: Why is it that some mountains look smaller than others but have larger altitudes? Task 2: There are four mountains named Annapurna I to Annapurna IV. How have the labels I to IV been assigned to these mountains?

38 38 The town of Pokhara has an altitude of 915 metres above sea level. Task 3: Find the height above Pokhara of each of the seven mountains and enter these heights into the table below. (Hiunchuli is given as a check.) Mountain Height above Pokhara Angle of Elevation from Pokhara Distance from Pokhara (km) Annapurna South 8.72 Annapurna I 8.37 Hiunchuli 5526m 8.15 Machhapuchre Annapurna III Annapurna IV Annapurna II The angle of elevation of the summit of each of the mountains has also been entered into the table. Task 4: Use the tangent ratio to find the distance from Pokhara of each of the seven mountains and enter these into the last column in the table. The calculation for Annapurna South has been started here.

39 39 Task 5: Use the original sketch of the mountains and your calculations of distance to place each of the seven mountains on this map. The span of the map roughly matches the width of the original sketch.

40 40 Application C: Train Tracks For a train track to operate successfully, the gradient of the track must be no more than one in forty, written as 1:40 or If the track has a steeper gradient, the trains will not be able to climb the track during wet or icy weather. The diagram here shows a gradient of 1:40. Task 1: Use the tangent ratio to find the maximum angle of incline (è) for a railway track. This is the angle between the track and horizontal. Task 2: Sketch a diagram showing the situation of a railway line rising 120m over a distance of 6.5km. Task 3: Calculate whether the line from task 2 will have an acceptable gradient. Task 4: A railway line is to be built between a port (at sea level) and a mine (728m above sea level). If the port and the mine are 46km apart calculate whether the gradient of the line will be acceptable.

41 41 Task 5: This map shows the positions and altitudes of eight towns along with proposed railway links between the towns. Find which of the proposed links are possible by direct route by calculating the angle of incline of the track using the altitudes and the given scale. Link Angle of Incline Possible / Not possible Algaville to Belaire Belaire to Cabrio Algaville to Delfmann Cabrio to Delfmann Cabrio to Erintown Cabrio to Fiore Delfmann to Greos Erintown to Fiore Fiore to Greos Greos to Hillman

42 42 Application D: Materials required: One piece of string measuring 12m, four marker pegs. Surveying In this application you are required to use some basic measuring equipment and your mathematical skills to mark out a rectangle as the base of a shed. Task 1: Devise a method to measure various lengths under 12m using your string. Task 2: Mark with pegs the corners of a rectangle that measures 4 metres by 7 metres. Ensure that the corner angles are 90 using techniques based on the mathematics you have learned in this module. Task 3: Get your teacher to measure the length of the diagonals of your rectangle. Task 4: Check the accuracy of your surveying by calculating the correct length of the diagonals. Task 5: Write a report explaining how you went about: Using the string to measure lengths Getting the side lengths correct Ensuring the 90 angles in the rectangle.

43 43 Application E: Project: Passive Solar Design In designing a passive solar home it is necessary to ensure that winter sun enters the house to warm it but that summer sun is kept out. This diagram shows how a well designed house can use the overhang of the roof to achieve this. Diagram A Task 1: Use the chart below to find the highest angle of elevation of the sun in summer and the lowest angle of the sun in winter for a house facing directly north (Orientation 000 ). Task 2: For what latitude does this chart apply?

44 44 Task 3: Using diagram A as a guide, draw an accurate scale diagram of the side of a North facing house which is 2.7m from the ground to the bottom of the roof. Place the window in a similar position to the one in diagram A. Show on your diagram that the summer sun at its highest will not enter the house but the winter sun will. You will need to create a roof overhang to the correct width to make this work. Ensure that the sun s angles are also correct. A north facing window on a home extension has no overhanging roof. To protect the window from direct sunlight an awning is to be placed over the window as in diagram B here: Diagram B Task 4: Using the measurements from diagram C below you have to design a rectangular awning as shown in diagram B that will cover the window from the summer sun but allow the winter sun to enter. Diagram C Task 5: Draw a scale drawing of the north end of the house showing where the awning is placed, its size and how it looks. Task 6: Draw a scale drawing of a side on view of the end of the house showing the awning and the suns rays in winter and summer.

45 45 Application F: Newspaper Study: Getting An Angle On It Task 1: This newspaper article tells of a bushfire that has a smoke plume visible from a distance. Find the angle of elevation of the top of the plume as viewed by someone in Ranville. Sydney (Tues) A bushfire rages out of control in the Gorman Valley north of Tillamen. The plume of smoke 2600 metres high was seen as far away as Ranville, 24 kilometres to the North. Task 2: These garden sheds have gable roofs with the last two dimensions giving the height (H) at the side and centre of the gable. Find the angle of elevation of the roof of each shed and the length of roof sheets required to be used on each of the sheds.

46 46 Task 3: Find the height of the ladder in this advertisement. LADDERS AT COST PRICE Safety standards state that a ladder of this type must have at least an angle of 40 in total between the angled sides. Does this ladder conform to requirements? New flexi ladders with a base spread of 2.2m and angle length of 3.4m available now! Task 4: These two excerpts from travel brochures each claim to be visiting the steepest pyramid. Which is actually steeper? Don t be fooled! Getupango travel gives you the steepest pyramid with a square base of 386m and a height of 236m. Gothere Travel Visit the world s steepest pyramid... base 456m square and 278m high.

47 47 Student Reflection What have I learned in this module? What new words did I learn during this module? Look at the outcomes at the start of the module (page 3). Have I progressed on each of these outcomes? What do I need to improve on? Write about one thing in this module I found interesting. What do I think was the most important concept in this module? Where could the maths in this module be used in our society? One area I would like to look more at is: Write something about how the bits in this module connect to each other. Write something about how the bits in this module connect to other modules.

48 48 ANSWERS TO EXERCISES: Exercise 1: 1. (a) 5, 60, 2.5, 8, 1, 20, 6 (b) 400, 600, 1.6, 2.5, 100, 20, 12.5 (c) 200, 400, 2.25, 275, 17.6, 0.056, (a) 1: (b) 1mm:2m (c) 1: (d) 1cm:50m (e) 1: (f) 1mm:200m (g) 1: (a) 400m (b) 2.5km (c) 125km (d) 1540km Exercise 2: 1. 64km You should be checking for approximate answers in the following questions. 2. (a) 280km (b) 165km (c) 725km (d) 830km km by 50km 4. 1 : (a) 610km (b) 570km (c) 1345km (d) 1765km km by 85km km Exercise 3: 1. 13cm mm m cm mm mm cm cm m mm mm Exercise 4: cm mm cm cm km m m or 139cm m m = 56.45, = He should not think it is square m and13.08m Exercise 5: cm m mm cm m mm mm cm mm cm Exercise 6: cm, 29.02cm, cm, 42.27, mm, 71.19mm, m, 10.86m, cm, 38.66, mm, 20.96mm, , 53.13, m Exercise 7: m m cm m 7. Ground distances 17.56m, 28.29m Lengths 40.05m, 64.53m cm