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FHSST Authors The Free high School Science texts : Textbooks for high School Students Studying the sciences Mathematics Grade 10. Version September 9, 2010. ii iii Copyright 2007 Free high School Science texts . Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version or any later version published by the Free Software Foundation; with no Invariant Sections, no Front- Cover texts , and no Back-Cover texts . A copy of the license is included in the section entitled GNU Free Documentation License . STOP!!!! Did you notice the FREEDOMS we've granted you? Our copyright license is different! It grants freedoms rather than just imposing restrictions like all those other Textbooks you probably own or use. We know people copy Textbooks illegally but we would LOVE it if you copied our's - go ahead copy to your hearts content, legally! Publishers revenue is generated by controlling the market, we don't want any money, go ahead, distribute our books far and wide - we DARE you!

FHSSTAuthors The Free High School Science Texts: Textbooks for High School Students Studying the Sciences Mathematics Grade 10 Version 0.5 September 9, 2010

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1 FHSST Authors The Free high School Science texts : Textbooks for high School Students Studying the sciences Mathematics Grade 10. Version September 9, 2010. ii iii Copyright 2007 Free high School Science texts . Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version or any later version published by the Free Software Foundation; with no Invariant Sections, no Front- Cover texts , and no Back-Cover texts . A copy of the license is included in the section entitled GNU Free Documentation License . STOP!!!! Did you notice the FREEDOMS we've granted you? Our copyright license is different! It grants freedoms rather than just imposing restrictions like all those other Textbooks you probably own or use. We know people copy Textbooks illegally but we would LOVE it if you copied our's - go ahead copy to your hearts content, legally! Publishers revenue is generated by controlling the market, we don't want any money, go ahead, distribute our books far and wide - we DARE you!

2 Ever wanted to change your textbook ? Of course you have! Go ahead change ours, make your own version, get your friends together, rip it apart and put it back together the way you like it. That's what we really want! Copy, modify, adapt, enhance, share, critique, adore, and contextualise. Do it all, do it with your colleagues, your friends or alone but get involved! Together we can overcome the challenges our complex and diverse country presents. So what is the catch? The only thing you can't do is take this book, make a few changes and then tell others that they can't do the same with your changes. It's share and share-alike and we know you'll agree that is only fair. These books were written by volunteers who want to help support education, who want the facts to be freely available for teachers to copy, adapt and re-use. Thousands of hours went into making them and they are a gift to everyone in the education community. iv v FHSST Core Team Mark Horner ; Samuel Halliday ; Sarah Blyth ; Rory Adams ; Spencer Wheaton FHSST Editors Jaynie Padayachee ; Joanne Boulle ; Diana Mulcahy ; Annette Nell ; Ren e Toerien ; Donovan Whitfield FHSST Contributors Sarah Abel ; Dr.

3 Rory Adams ; Andrea Africa ; Ben Anhalt ; Prashant Arora ; Raymond Barbour ; Richard Baxter ; Tara Beckerling ; Tim van Beek ; Jennifer de Beyer ; Dr. Sarah Blyth ; Sebastian Bodenstein ; Martin Bongers ; Stephan Brandt ; Craig Brown ; Graeme Broster ; Deanne de Bude ; Richard Case ; Fanny Cherblanc ; Dr. Christine Chung ; Brett Cocks ; Andrew Craig ; Tim Crombie ; Dan Crytser ; Dr. Anne Dabrowski ; Laura Daniels ;. Sean Dobbs ; Esmi Dreyer ; Matthew Duddy ; Fernando Durrell ; Dr. Dan Dwyer ; Frans van Eeden ; Alex Ellis ; Tom Ellis ; Giovanni Franzoni ; Ingrid von Glehn ; Tamara von Glehn ;. Lindsay Glesener ; Kevin Godby ; Dr. Vanessa Godfrey ; Dr. Johan Gonzalez ; Hemant Gopal ;. Dr. Stephanie Gould ; Umeshree Govender ; Heather Gray ; Lynn Greeff ; Dr. Tom Gutierrez ;. Brooke Haag ; Kate Hadley ; Dr. Sam Halliday ; Asheena Hanuman ; Dr Melanie Dymond Harper ; Dr. Nicholas Harrison ; Neil Hart ; Nicholas Hatcher ; Dr. William P.

4 Heal ; Pierre van Heerden ; Dr. Fritha Hennessy ; Millie Hilgart ; Chris Holdsworth ; Dr. Benne Holwerda ;. Dr. Mark Horner ; Mfandaidza Hove ; Robert Hovden ; Jennifer Hsieh ; Clare Johnson ; Luke Jordan ; Tana Joseph ; Dr. Fabian Jutz ; Dr. Lutz Kampmann ; Paul Kim ; Dr. Jennifer Klay ;. Lara Kruger ; Sihle Kubheka ; Andrew Kubik ; Dr. Jannie Leach ; Dr. Marco van Leeuwen ;. Dr. Tom Leinster ; Dr. Anton Machacek ; Dr. Komal Maheshwari ; Kosma von Maltitz ;. Bryony Martin ; Nicole Masureik ; John Mathew ; Dr. Will Matthews ; JoEllen McBride ;. Nikolai Meures ; Riana Meyer ; Filippo Miatto ; Jenny Miller ; Abdul Mirza ; Mapholo Modise ;. Carla Moerdyk ; Asogan Moodaly ; Jothi Moodley ; David Myburgh ; Kamie Naidu ; Nolene Naidu ; Bridget Nash ; Tyrone Negus ; Thomas O'Donnell ; Dr. Markus Oldenburg ; Dr. Jaynie Padayachee ; Dave Pawson ; Nicolette Pekeur ; Sirika Pillay ; Jacques Plaut ; Andrea Prinsloo ; Joseph Raimondo ; Sanya Rajani ; Prof.

5 Sergey Rakityansky ; Alastair Ramlakan ;. Dr. Matina J. Rassias ; Dr. Jocelyn Read ; Dr. Matthew Reece ; Razvan Remsing ; Laura Richter ; Max Richter ; Sean Riddle ; Jonathan Reader ; Dr. David Roberts ; Evan Robinson ;. Raoul Rontsch ; Dr. Andrew Rose ; Katie Ross ; Jeanne-Mari e Roux ; Bianca Ruddy ; Katie Russell ; Steven Sam ; Nathaniel Schwartz ; Duncan Scott ; Helen Seals ; Ian Sherratt ; Dr. James Short ; Roger Sieloff ; Clare Slotow ; Bradley Smith ; Greg Solomon ; Dr. Andrew Stacey ; Dr. Jim Stasheff ; Mike Stay ; Mike Stringer ; Tim Teatro ; Ben Thompson ; Shen Tian ; Nicola du Toit ; Robert Torregrosa ; Jimmy Tseng ; Pieter Vergeer ; Helen Waugh ; Dr. Dawn Webber ; Michelle Wen ; Neels van der Westhuizen ; Dr. Alexander Wetzler ; Dr. Spencer Wheaton ; Vivian White ; Dr. Gerald Wigger ; Harry Wiggins ; Heather Williams ;. Wendy Williams ; Julie Wilson ; Timothy Wilson ; Andrew Wood ; Emma Wormauld ; Dr. Sahal Yacoob ; Jean Youssef ; Ewald Zietsman Contributors and editors have made a sincere effort to produce an accurate and useful resource.

6 Should you have suggestions, find mistakes or be prepared to donate material for inclusion, please don't hesitate to contact us. We intend to work with all who are willing to help make this a continuously evolving resource! vi Contents I Basics 1. 1 Introduction to Book 3. The Language of Mathematics .. 3. II Grade 10 5. 2 Review of Past Work 7. Introduction .. 7. What is a number? .. 7. Sets .. 7. Letters and Arithmetic .. 8. Addition and Subtraction .. 9. Multiplication and Division .. 9. Brackets .. 9. Negative Numbers .. 10. What is a negative number? .. 10. Working with Negative Numbers .. 11. Living Without the Number Line .. 12. Rearranging Equations .. 13. Fractions and Decimal Numbers .. 15. Scientific Notation .. 15. Real Numbers .. 16. Natural Numbers .. 17. Integers .. 17. Rational Numbers .. 17. Irrational Numbers .. 19. Mathematical Symbols .. 20. Infinity .. 20. End of Chapter Exercises .. 20. 3 Rational Numbers - Grade 10 23.

7 Introduction .. 23. The Big Picture of Numbers .. 23. Definition .. 24. vii CONTENTS CONTENTS. Forms of Rational Numbers .. 25. Converting Terminating Decimals into Rational Numbers .. 25. Converting Repeating Decimals into Rational Numbers .. 26. Summary .. 27. End of Chapter Exercises .. 27. 4 Exponentials - Grade 10 29. Introduction .. 29. Definition .. 29. Laws of Exponents .. 30. Exponential Law 1: a0 = 1 .. 30. Exponential Law 2: am an = am+n .. 30. 1. Exponential Law 3: a n = an , a 6= 0 .. 31. m n m n Exponential Law 4: a a = a .. 32. Exponential Law 5: (ab)n = an bn .. 32. Exponential Law 6: (am )n = amn .. 33. End of Chapter Exercises .. 34. 5 Estimating Surds - Grade 10 37. Introduction .. 37. Drawing Surds on the Number Line (Optional) .. 38. End of Chapter Exercises .. 39. 6 Irrational Numbers and Rounding Off - Grade 10 41. Introduction .. 41. Irrational Numbers .. 41. Rounding Off .. 42. End of Chapter Exercises.

8 43. 7 Number Patterns - Grade 10 45. Common Number Patterns .. 45. Special Sequences .. 46. Make your own Number Patterns .. 46. Notation .. 47. Patterns and Conjecture .. 49. Exercises .. 50. 8 Finance - Grade 10 53. Introduction .. 53. Foreign Exchange Rates .. 53. How much is R1 really worth? .. 53. Cross Currency Exchange Rates .. 56. Enrichment: Fluctuating exchange rates .. 57. Being Interested in Interest .. 58. viii CONTENTS CONTENTS. Simple Interest .. 59. Other Applications of the Simple Interest Formula .. 62. Compound Interest .. 64. Fractions add up to the Whole .. 65. The Power of Compound Interest .. 66. Other Applications of Compound Growth .. 67. Summary .. 69. Definitions .. 69. Equations .. 69. End of Chapter Exercises .. 69. 9 Products and Factors - Grade 10 71. Introduction .. 71. Recap of Earlier Work .. 71. Parts of an Expression .. 71. Product of Two Binomials .. 71. Factorisation .. 72. More Products.

9 74. Factorising a Quadratic .. 76. Factorisation by Grouping .. 79. Simplification of Fractions .. 80. End of Chapter Exercises .. 82. 10 Equations and Inequalities - Grade 10 83. Strategy for Solving Equations .. 83. Solving Linear Equations .. 84. Solving Quadratic Equations .. 89. Exponential Equations of the Form ka(x+p) = m .. 94. Algebraic Solution .. 94. Linear Inequalities .. 97. Linear Simultaneous Equations .. 100. Finding solutions .. 100. Graphical Solution .. 100. Solution by Substitution .. 102. Mathematical Models .. 104. Introduction .. 104. Problem Solving Strategy .. 105. Application of Mathematical Modelling .. 105. End of Chapter Exercises .. 107. 11 Functions and Graphs - Grade 10 109. Introduction to Functions and Graphs .. 109. Functions and Graphs in the Real-World .. 109. ix CONTENTS CONTENTS. Recap .. 110. Variables and Constants .. 110. Relations and Functions .. 110. The Cartesian Plane .. 111.

10 Drawing Graphs .. 111. Notation used for Functions .. 112. Characteristics of Functions - All Grades .. 114. Dependent and Independent Variables .. 115. Domain and Range .. 115. Intercepts with the Axes .. 115. Turning Points .. 116. Asymptotes .. 116. Lines of Symmetry .. 116. Intervals on which the Function Increases/Decreases .. 116. Discrete or Continuous Nature of the Graph .. 117. Graphs of Functions .. 118. Functions of the form y = ax + q .. 118. Functions of the Form y = ax2 + q .. 123. a Functions of the Form y = x + q .. 128. Functions of the Form y = ab(x) + q .. 132. End of Chapter Exercises .. 136. 12 Average Gradient - Grade 10 Extension 137. Introduction .. 137. Straight-Line Functions .. 137. Parabolic Functions .. 138. End of Chapter Exercises .. 139. 13 Geometry Basics 141. Introduction .. 141. Points and Lines .. 141. Angles .. 142. Measuring angles .. 142. Special Angles .. 143. Special Angle Pairs .. 145.


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