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MYP Mathematics PDF

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MYP Mathematics A concept-based approach 4&5 Extended Rose Harrison • Clara Huizink Aidan Sproat-Clements • Marlene Torres-Skoumal 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. p215: photopixel / Shutterstock; p216: szefei / Shutterstock; p217: It furthers the University’s objective of excellence in research, Amra Pasic / Shutterstock; p233: Teresa Levite / Shutterstock; p248: scholarship, and education by publishing worldwide. Oxford is a 3Dsculptor / Shutterstock; p249: Dmitry Pichugin / Shutterstock; registered trade mark of Oxford University Press in the UK and in p264: Podfoto / Shutterstock; p279: © The Carlisle Kid; p279: certain other countries. Tyler Olson / Shutterstock; p281: mama_mia / Shutterstock; p281: Joel_420 / Shutterstock; p282: noolwlee / Shutterstock; p286: © Oxford University Press 2017 Maks Narodenko / Shutterstock; p296: Monkey Business Images / The moral rights of the authors have been asserted. Shutterstock; p299: Daxiao Productions / Shutterstock p306: Tina First published in 2017 Renceij/123rf. All rights reserved. No part of this publication may be reproduced, Although we have made every effort to trace and contact all stored in a retrieval system, or transmitted, in any form or by copyright holders before publication this has not been possible any means, without the prior permission in writing of Oxford in all cases. If notied, the publisher will rectify any errors or University Press, or as expressly permitted by law, by licence or omissions at the earliest opportunity. under terms agreed with the appropriate reprographics rights Links to third party websites are provided by Oxford in good faith organization. Enquiries concerning reproduction outside the scope and for information only. Oxford disclaims any responsibility for of the above should be sent to the Rights Department, Oxford the materials contained in any third party website referenced in University Press, at the address above. this work. You must not circulate this work in any other form and you must impose this same condition on any acquirer. British Library Cataloguing in Publication Data Data available 978-0-19-835619-6 1 3 5 7 9 10 8 6 4 2 Paper used in the production of this book is a natural, recyclable product made from wood grown in sustainable forests. The manufacturing process conforms to the environmental regulations of the country of origin. Printed in Great Britain by Bell and Bain Ltd. Glasgow. Acknowledgements The publishers would like to thank the following for permissions to use their photographs/illustrations: Cover image: NASA; p2: MarcelClemens / Shutterstock; p3: Eric Isselee / Shutterstock; p6: wanchai / Shutterstock; p8: Jon Mackay; p11: Everett Historical / Shutterstock; p12: Wollertz / Shutterstock; p14: Richard Peterson / Shutterstock; p16: asharkyu / Shutterstock; p17: Jon Mackay; p20: dedek / Shutterstock; p21: RTimages / Shutterstock; p29: Sergey Dubrov / Shutterstock; p30: George Dolgikh / Shutterstock; p32: jeff Metzger / Shutterstock; p35: BortN66 / Shutterstock; p48: Andresr / Shutterstock; p54: Scott David Patterson / Shutterstock; p71: perspectivestock / Shutterstock; p76: PEPPERSMINT / Shutterstock; p80: sisqopote / Shutterstock; p85: Jacek Chabraszewski / Shutterstock; p91: Dionisvera / Shutterstock; p92: Boris Sosnovyy / Shutterstock; p93: Eric Isselee / Shutterstock; p94: Monkey Business Images / Shutterstock; p95: Jonathan Lenz / Shutterstock; p96: LeventeGyori / Shutterstock; p98: Stephen Finn / Shutterstock; p108: Ales Liska / Shutterstock; p113: Eugene Sim / Shutterstock; p116: visceralimage / Shutterstock; p117: wanchai / Shutterstock; p119: Seaphotoart / Shutterstock; p120: injun / Shutterstock; p120: Marco Tomasini / Shutterstock; p123: Christian Delbert / Shutterstock; p124: bunnyphoto / Shutterstock; p127: ILYA AKINSHIN / Shutterstock; p127: photka / Shutterstock; p128: AVS-Images / Shutterstock; p140: EpicStockMedia / Shutterstock; p144: SKA Organisation; p145: Pinosub / Shutterstock; p157: Jonathan Noden-Wilkinson / Shutterstock; p162: greenland / Shutterstock; p166: Andrey Pavlov / Shutterstock; p167: edobric / Shutterstock; p168: Jon Mackay; p171: SmileStudio / Shutterstock; p175: Jon Mackay; p182: IM_photo / Shutterstock; p183: Gail Johnson / Shutterstock; p197: Jon Mackay; p198: Dmitry Kalinovsky / Shutterstock; p200: Antiqua Print Gallery / Alamy Stock Photo; p202: photopixel / Shutterstock; p203: Paul André Belle-Isle / Shutterstock; p204: nuttakit / Shutterstock; p205: hironai / Shutterstock; p207: Konstantin Yolshin / Shutterstock; How to use this book Chapters Chapters 1–3 each focus on one of the key concepts: Form, Relationships and Logic. Chapters 4–15 each focus on one of the twelve related concepts for Mathematics: Representation, Simplication, Quantity, Measurement, Patterns, Space, Change, Equivalence, Generalization, Justication, Models, and Systems Creating your own units The suggested unit structure opposite shows just one way of grouping all the topics from different chapters, from both Standard and Extended, to create units. These units have been created following the ofcial guidelines from the IB Building Quality Curriculum. Each unit is driven by a meaningful statement of inquiry and is set within a relevant global context. Each chapter focuses on one of the IB’s twelve related concepts for ● Mathematics, and each topic focuses on one of the three key concepts. Hence, these units, combining topics from different chapters, connect the key and related concepts to help students both understand and remember them. Stand-alone topics in each chapter teach mathematical skills and how ● to apply them, through inquiry into factual, conceptual, and debatable questions related to a global context. This extends students’ understanding and ability to apply mathematics in a range of situations. You can group topics as you choose, to create units driven by a ● contextualized statement of inquiry. This book covers the MYP Extended skills framework. ● MYP Mathematics 4 & 5Standard covers the Standard skills. Using MYP Mathematics 4 & 5 Extended with an existing scheme of work If your school has already established units, statements of inquiry and global contexts, you can easily integrate the concept-based topics in this book into your current scheme of work. The table of contents on page vii clearly shows the topics covered in each concept-based chapter. Your scheme’s units may assign a different concept to a given topic than we have. In this case, you can simply add the concept from this book to your unit plan. Most topics include a Review in context, which may differ from the global context chosen in your scheme of work. In this case, you may wish to write some of your own review questions for your global context, and use the questions in the book for practice in applying mathematics in different scenarios. ii Suggested plan The units here have been put together by the Access your support website for more authors as just one possible way to progress suggested plans for structuring units: through the content. www.oxfordsecondary.com/myp-mathematics Units 1, 5, 6, 9 and 10 are made up of topics solely from the MYPMathematics 4 & 5 Standard book. UNIT 2 UNIT 11 43 123 E41 73 91 E91 Topics: Scatter graphs and linear regression, Topics: Circle segments and sectors, volumes drawing reasonable conclusions, data inferences of 3D shapes, 3D orientation Global context: Identities and relationships Global context: Personal and cultural expression Key concept: Relationships Key concept: Relationships UNIT 3 UNIT 12 111 143 E141 131 132 E131 Topics: Equivalence transformations, inequalities, Topics: Using circle theorems, intersecting chords, non-linear inequalities problems involving triangles Global context: Identities and relationships Global context: Personal and cultural expression Key concept: Form Key concept : Logic UNIT 4 UNIT 13 51 103 E51 102 113 E102 Topics: Rational and irrational numbers, direct Topics: Algebraic fractions, equivalent methods, and indirect proportion, fractional exponents rational functions Global context: Globalization and sustainability Global context: Scientic and technical innovation Key concept: Form Key concept: Form UNIT 7 UNIT 14 81 121 122 E81 E61 E101 E121 Topics: Finding patterns in sequences, making Topics: Evaluating logarithms, transforming generalizations from a given pattern, arithmetic logarithmic functions, laws of logarithms and geometric sequences Global context: Orientation in space and time Global context: Scientic and technical innovation Key concept: Relationships Key concept: Form UNIT 15 UNIT 8 E71 E92 E111 44 151 E151 Topics: The unit circle and trigonometric functions, Topics: Simple probability, probability systems, sine and cosine rules, simple trigonometric conditional probability identities Global context: Identities and relationships Global context: Orientation in space and time Key concept: Logic Key concept: Relationships iii About the authors Marlene Torres-Skoumal has taught DP Rose Harrison is the Lead Educator for MYP and MYP Mathematics for several decades. Mathematics at the IB Organization. Sheis an In addition to being a former Deputy Chief MYP and DP workshop leader, a senior reviewer Examiner for IB HL Mathematics, she is an for Building Quality Curriculum in the MYP, and IB workshop leader, and has been a member has held many positions of responsibility in her of various curriculum review teams. Marlene 20 years’ experience teaching Mathematics in hasauthored both DP and MYP books, international schools. including Higher Level Mathematics for Aidan Sproat-Clements is Head of Mathematics OxfordUniversity Press. at Wellington College in the UK, an IB World Clara Huizink has taught MYP and DP School. He has spent his career in British Mathematics at international schools in the independent schools where he has promoted Philippines, Austria and Belgium. She has rigorous student-led approaches to the learning alsobeen through the IB experience herself of Mathematics. He helped to develop the pilot as a student and she is a graduate of the MYP eAssessments for Mathematics. IBprogram. Topic opening page Global context – some questions in Objectives – the E41 How to stand out from this unit are set in mathematics this context. You the crowd covered in this topic. may wish to write Global context: Identities and relationships your own global Objectives Inquiry questions Making inferences about data, given the mean F context to engage Key concept for and standard deviation deviation of a data set? Using dierent forms of the standard deviation your students. the topic. formula C How is the meaning of ‘standard S Understanding the normal distribution SHIP MUsaiknign tgh ien sfetarenndcaersd adbeovuiat tnioonrm anald d tihsetr mibueatinons D fCoarmn sualamsp?les give reliable results? N Using unbiased estimators of the population Do we want to be like everybody else? O mean and standard deviation Inquiry questions– TI A ATL – the Approach EL the factual, R to Learning taught conceptual, and ATL Communication in this topic. Make inferences and draw conclusions debatable questions explored in this 43 topic. 123 Unit plan – shows E41 Statement of the topics in the Inquiry for this Standard and unit. You may wish Statement of Inquiry: Extended books to write your own. Generalizing and representing relationships that you could can help to clarify trends among individuals. teach together as 76 a unit. iv Learning features Each topic has three sections, exploring: factual ● conceptual ● debatable inquiry questions ● Problem solving – where the method of solution is not immediately obvious, these are highlighted in the Practices. Explorations are inquiry-based learning activities for students working inidvidually, in pairs or in small groups to discover mathematical facts and concepts. Exploration 4 ATL Using dynamic geometry software, plot the points A (10, 6) and B (2, 8). M, the midpoint of AB 3 Compare the coordinates of M A and B Describe anything you notice. 4 Create other pairs of points and nd the midpoint of each pair. Investigate the relationship between the coordinates of the endpoints ATL highlights an opportunity to develop the and the coordinates of the midpoint. (Hint: construct a table of endpoints and midpoints.) ATL skill identied on the topic opening page. 5 Use your ndings in step 4 to predict the coordinates of N, the midpoint of points C(7, 11) and D (19, 15). 6 Verify your answer using the software. 7 Points P and Q have coordinates P ) and Q d). Suggest a formula for the coordinates of the midpoint of PQ 8 Verify that your formula gives the correct coordinates for M and N Reect and discuss – opportunities for small Reect and discuss 3 group or whole class reection and discussion Generalization means making a general statement on the basis of specic examples. Where in Exploration 4 have you generalized? on their learning and the inquiry questions. How does generalization enable you to predict? Why is it important to verify a generalization? The midpoint of ( ) and ( d) is . Worked examples show a clear solution and Example 1 explain the method. Find the midpoint of A(3, 6) and B (9, 18). Let M be the midpoint of AB M 3+96+18 Use the midpoint formula M = (6, 12) Practice 3 Practice questions written using IB command 1 Find the midpoint of A(14, 6) and B (18, 20). terms, to practice the skills taught and how to 2 Find the midpoint of C (5, 13) and D (10, 8). 3 a Plot the points A(2, 7), B (3, 10), C (6, 11), and D (5, 8). apply them to unfamiliar problems. b Find the midpoints of AC and BD Comment on your answers to part b. Explain what this shows about quadrilateral ABCD 70 3 Logic Objective boxes highlight an IB Assessment Objective and explain to students how to satisfy the objective in an Exploration or Practice. Technology icon Using technology allows students to discover new ideas through examining a wider range of examples, or to access complex ideas without having to do lots of painstaking work by hand. This icon shows where students could use Graphical Display Calculators (GDC), Dynamic Geometry Software (DGS) or Computer Algebra Systems (CAS). Places where students should not use technology are indicated with a crossed-out icon. The notation used throughout this book is largely that required in the DP IB programs. v Each topic ends with: Summary The standard deviation is a measure of Using notation, the mean of a set of discrete dispersion that gives an idea of how close the data values in a frequency table is , the sum original data values are to the mean, and so how representative the mean is of the data. of all the x f values divided by the total frequency. A small standard deviation shows that the A formula to calculate the standard deviation for data values are close to the mean. The units of discrete data presented in a frequency table is: Summary of the standard deviation are the same as the units of the original data. key points x means the sum of all the x values. Using this notation, the mean of a set of values The normal distribution is a symmetric is Σx, the sum of all the values divided by the distribution, with most values close to the mean n and tailing o evenly in either direction. Its number of values. frequency graph is a bell-shaped curve. Mixed practice Number of strawberries on each plant fed with In questions 1 and 2 the data provided is for special strawberry plant food: Mixed practice – the population. 14 15 17 17 19 19 12 14 15 15 summative 1 Findthe mean and the standard deviation of Analyze the data to nd whether there is an assessment of the each data set: eect of using the special strawberry plant food. facts and skills a 2, 3, 3, 4, 4, 5, 5, 6, 6, 6 3 Bernie recorded the weight of food in grams his b 21 kg, 21 kg, 24 kg, 25 kg, 27 kg, 29 kg hamster ate each day. Here are his results: learned, including c problem-solving x f Day 1 2 3 4 5 6 7 8 9 questions, and 3 2 Food (g) 55 64 45 54 60 50 59 61 49 4 3 a Find the mean and standard deviation of questions in a 5 2 the weight of food the hamster ate over the range of contexts. 9days. d Interval Frequency b Bernie assumes that his hamster’s food − Review in context The ancient art of origami has roots in Japanese The resulting creases are illustrated in this diagram. culture and involves folding at sheets of paper A B C to create models. Some models are designed to resemble animals, owers or buildings; others E just celebrate the beauty of geometric design. Review in context – summative D F assessment questions within the global H context for the topic. G Z The artist thinks lengths FH and DY are equal. b Explain why EX 1EG. 2 Reect and discuss Reection on the statement of How have you explored the statement of inquiry? Give specic examples. inquiry vi Table of contents Access your support website: www.oxfordsecondary.com/myp-mathematics Form Space E1.1 W hat if we all had eight ngers? • E9.1 A nother dimension • Dierent bases 2 3D Orientation 167 E9.2 M apping the world • Relationships Oblique triangles 183 E2.1 M ultiple doorways • Composite functions 21 Change E2.2 D oing and undoing • E10.1 Time for a change • Invserse functions 35 Log functions 202 E10.2 M eet the transformers • Logic Rational functions 216 E3.1 H ow can we get there? • Vectors and vector spaces 54 Equivalence E11.1 U nmistaken identities • Representation Trigonometric identities 233 E4.1 H ow to stand out from the crowd • Data inferences 76 Generalization E12.1 G o ahead and log in • Simplication Laws of logarithms 249 E5.1 Super powers • Fractional exponents 98 Justication E13.1 A re we very similar? • Quantity Problems involving triangles 264 E6.1 I deal work for lumberjacks • Evaluating logs 108 Models E14.1 A world of dierence • Measurement Nonlinear inequalities 282 E7.1 S lices of pi • The unit circle and Systems trignometric functions E15.1 B ranching out • 124 Conditional probability 299 Patterns Answers 315 E8.1 M aking it all add up • Arithmetic and geometric sequences 145 Index 350 vii E11 What if we all had eight . ngers? Global context: Scientic and technical innovation Objectives Inquiry questions ● Understanding the concept of a number F ● How have numbers been written in system history? Counting in dierent bases What is a number base? ● ● Converting numbers from one base to How can you write numbers in other ● ● M another bases? R Using operations in dierent bases O ● C ● How are mathematical operations in other bases similar to and dierent from F operations in base 10? D ● Would you be better o counting in base 2? How does form inuence function? ● ATL Communication Use intercultural understanding to interpret communication 2 NUMBER You should already know how to: use the operations of addition, 1 Calculate these by hand: ● subtraction and long multiplication a 10442 + 762 b 10887 7891 in base 10, without a calculator c 27 × 43 d 14078 × 71 understand place value 2 Write down the value of the 5 in: ● a 351 b 511002 c 15 d 1.5 F Numbers in dierent bases How have numbers been written in history? ● What is a number base? ● How can you write numbers in other bases? ● Humans have used many dierent ways to record numbers. How would you write down the number of green bugs in this diagram? You might have written a symbol 5, or the word “ve”. You could have used a word in a dierent language, or maybe even a tally: . Each of these represents the number in a dierent way, but they all represent the same quantity. Dierent cultures use dierent forms of notation to represent number. The ancient Egyptian hieroglyphic number system was an additive system. Each symbol has a dierent value and you nd the total value of the number by adding the values of all the symbols together. Egyptian numerals use these symbols: Green shield bugs are sometimes called green stink stroke heelbone coiled rope bugs, as they 1 10 100 produce a pungent odor if handled or disturbed. lotus ower pointed nger tadpole scribe 1000 10000 100000 1000000 The number 11 is written , and 36 is written . E1.1 What if we all had eight ngers? 3

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MYP Mathematics. A concept-based approach. 4&5. Extended. Rose Harrison • Clara Huizink. Aidan Sproat-Clements • Marlene Torres-Skoumal . Topics: Algebraic fractions, equivalent methods, rational functions. Global context: Scienti c and technical innovation. Key concept: Form. UNIT 14. E6 1.
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Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.