7. Coordinate geometry
16. Exponential and trigonometric functions
18. Two variable statistics
19. Introductory differential calculus
Mathematics for the International Student: Mathematical Studies SL (International Baccalaureate Diploma Programme) is our interpretation of the concepts outlined in the IBO Diploma Programme Mathematical Studies SL Guide (first examinations 2006). It is not our intention to define the course, and teachers are encouraged to use other resources. We have developed the book independently of the International Baccalaureate Organization (IBO) in consultation with many experienced teachers of IB Mathematics, but the text is not endorsed by the IBO.
Our package is language rich and technology rich, comprising a textbook with an interactive CD. The combination of textbook and interactive Student CD will foster the mathematical development of students in a stimulating way. Frequent use of the interactive features on the CD is certain to nurture a much deeper understanding and appreciation of mathematical concepts.
The CD displays the contents of the textbook with many interactive features such as simulations, computer demonstrations, video clips, graphing packages, spreadsheets etc. to stimulate the interest of students and assist teachers.
The book contains many problems, from basic to the advanced, to cater for a wide range of student abilities and interests. Some exercises are simply designed to build skills, but every effort is made to contextualise problems to help students see the everyday uses and practical application of the mathematics they are studying, and appreciate the universality of mathematics.
Emphasis is placed on the gradual development of concepts with appropriate worked examples and we have also provided Investigations and extension material to engage and challenge more capable students.
By offering one book for Mathematical Studies SL, teachers have the flexibility to teach the 2-year course in whatever order they choose. For those who would like some guidance, it is suggested that students work progressively from Chapter 1 through to Chapter 11 for the first year, although some teachers may prefer to include Chapter 18 ‘Two-variable statistics’ in the first year as a basis for internal assessment. Other teachers may prefer to leave statistics until early in the second year and then have students work progressively from Chapter 12 through to Chapter 19.
However it is acknowledged that there is no single best way for all teachers to work through the syllabus. Individual teachers have to consider particular needs of their students and other requirements and preferences that they may have. We invite teachers to email their preferred order or suggestions to us. We can put these suggestions on our website to be shared with other teachers.
The extensive use of graphics calculators and computer packages throughout our books enables students to appreciate the importance, application and appropriate use of technology. No single aspect of technology has been favoured. It is as important that students work with a pen and paper as it is that they use their calculator or graphics calculator, or use a spreadsheet or graphing package on computer.
In this changing world of mathematics education, we believe the contextual approach shown in this book, with the associated use of technology, will enhance the students’ understanding, knowledge and appreciation of mathematics.
The textbook is accompanied by an interactive Student CD (which is tucked in a plastic wallet on the inside back cover of each textbook). The CD includes the complete text of the book and about 150 interactive links.
The interactive features of the CD allow immediate access to our own specially designed geometry packages, graphing packages and more. Teachers are provided with a quick and easy method of demonstrating concepts, and students can explore for themselves, and revisit when necessary.
Simply ‘click’ the active icon to access a range of interactive features:
For a complete list of all the active links on the Mathematical Studies SL CD, click here.
The CD is ideal for independent study. Frequent use is sure to nurture a deeper understanding of Mathematics. Students can revisit concepts taught in class and undertake their own revision and practice. The CD also has the text of the book, permitting students to leave the textbook at school and keep the CD at home.
Each author is a highly experienced teacher of mathematics at senior school level and Mal Coad has been a senior examiner & moderator for IB Diploma Mathematical Studies for many years. Haese & Harris Publications is a specialist publisher of quality mathematics textbooks and software. Please email Ray O’Farrell. We welcome your feedback.
Email: ray@haeseandharris.com.au
| 1 | NUMBER SETS AND PROPERTIES | 17 | |
| A | Some set language | 18 | |
| B | Number sets | 19 | |
| C | Words used in mathematics | 22 | |
| D | Exponential (index) notation | 24 | |
| E | Factors of natural numbers | 25 | |
| Investigation: The sieve of Eratosthenes | 28 | ||
| F | Multiples of natural numbers | 29 | |
| G | Order of operations | 31 | |
| Review set 1A | 33 | ||
| Review set 1B | 34 | ||
| 2 | MEASUREMENT | 35 | |
| A | Time | 36 | |
| B | Temperature | 39 | |
| C | Imperial standard units | 40 | |
| D | Standard form (scientific notation) | 42 | |
| E | Rounding numbers | 45 | |
| F | Rates | 51 | |
| Investigation 1: Stopping distances | 52 | ||
| G | Measuring devices and their accuracy | 56 | |
| Investigation 2: Measuring devices | 56 | ||
| H | Accuracy of measurements | 58 | |
| I | Error and percentage error | 61 | |
| Investigation 3: A gram in the hand is worth… | 61 | ||
| Investigation 4: Estimating and accuracy | 63 | ||
| Review set 2A | 64 | ||
| Review set 2B | 65 | ||
| 3 | SETS AND VENN DIAGRAMS | 67 | |
| A | Set builder notation | 69 | |
| B | Complements of sets | 70 | |
| C | Venn diagrams | 73 | |
| D | Venn diagram regions | 76 | |
| E | Numbers in regions | 77 | |
| Review set 3A | 82 | ||
| Review set 3B | 83 | ||
| 4 | THE RULE OF PYTHAGORAS | 85 | |
| A | The rule of Pythagoras (review) | 86 | |
| B | Pythagoras and geometrical figures | 91 | |
| C | The converse of Pythagoras’ rule | 93 | |
| Investigation 1: Pythagorean triples spreadsheet | 94 | ||
| D | Problem solving | 95 | |
| E | True bearings and navigation | 99 | |
| Investigation 2: Shortest distance | 101 | ||
| F | Circle problems | 102 | |
| G | Three-dimensional problems | 104 | |
| Review set 4A | 106 | ||
| Review set 4B | 107 | ||
| Review set 4C | 108 | ||
| 5 | DESCRIPTIVE STATISTICS | 109 | |
| A | Describing data | 110 | |
| Investigation 1: Statistics from the Internet | 112 | ||
| B | Presenting and interpreting data | 112 | |
| C | Grouped discrete data | 117 | |
| Investigation 2: Taxi Sir? | 119 | ||
| D | Continuous data | 121 | |
| Investigation 3: Choosing class intervals | 122 | ||
| E | Frequency distribution tables | 123 | |
| F | Summarising the data | 128 | |
| Investigation 4: Effects of outliers | 140 | ||
| G | Measuring the spread of data | 141 | |
| H | Box-and-whisker plots | 146 | |
| I | The standard deviation | 149 | |
| Investigation 5: Heart stopper | 154 | ||
| J | Statistics using technology | 155 | |
| K | Parallel boxplots | 158 | |
| Investigation 6: How do you like your eggs? | 160 | ||
| Review set 5A | 161 | ||
| Review set 5B | 163 | ||
| 6 | LINEAR AND EXPONENTIAL ALGEBRA | 165 | |
| A | Algebraic substitution | 167 | |
| Investigation 1: Solving equations | 168 | ||
| B | Linear equations | 169 | |
| C | Fractional equations | 172 | |
| D | Problem solving | 175 | |
| E | Formula substitution | 177 | |
| F | Formula rearrangement | 180 | |
| Investigation 2: The cycles problem | 181 | ||
| G | Linear simultaneous equations | 182 | |
| H | Problem solving | 187 | |
| I | Index notation (Review) | 189 | |
| J | Negative bases | 190 | |
| K | Index laws | 192 | |
| L | Exponential equations | 197 | |
| Review set 6A | 199 | ||
| Review set 6B | 200 | ||
| Review set 6C | 201 | ||
| 7 | COORDINATE GEOMETRY | 203 | |
| A | Distance between two points | 205 | |
| B | Gradient | 207 | |
| C | Applications of gradient | 213 | |
| D | Midpoints | 215 | |
| E | Vertical and horizontal lines | 218 | |
| F | Equations of lines | 219 | |
| G | Graphing lines | 226 | |
| Investigation: Finding where lines meet using technology | 228 | ||
| H | Midpoints and perpendicular bisectors | 229 | |
| Review set 7A | 232 | ||
| Review set 7B | 233 | ||
| Review set 7C | 234 | ||
| 8 | QUADRATIC ALGEBRA | 235 | |
| A | Products and expansions | 236 | |
| Investigation 1: The product of three consecutive integers | 241 | ||
| B | Further expansion | 243 | |
| Investigation 2: The expansion of (a + b)3 | 245 | ||
| C | Factorisation of quadratic expressions | 245 | |
| D | Factorisation of ax2 + bx + c (a ≠ 1) | 252 | |
| E | Quadratic equations | 255 | |
| F | Completing the square | 259 | |
| Investigation 3: The quadratic formula (Extension) | 262 | ||
| G | Problem solving with quadratics | 263 | |
| Review set 8A | 265 | ||
| Review set 8B | 265 | ||
| Review set 8C | 266 | ||
| 9 | FUNCTION NOTATION AND QUADRATIC FUNCTIONS | 267 | |
| A | Relations and functions | 268 | |
| B | Interval notation, domain and range | 271 | |
| C | Function notation | 274 | |
| D | Functions as mappings | 276 | |
| Investigation 1: Fluid filling functions | 279 | ||
| E | Linear functions | 281 | |
| F | Quadratic functions | 286 | |
| G | Graphs of quadratic functions | 289 | |
| Investigation 2: Graphs of the form y = x2 + k | 291 | ||
| Investigation 3: Graphs of the form y = (x – h)2 | 292 | ||
| Investigation 4: Graphs of the form y = (x – h)2 + k | 293 | ||
| Investigation 5: Graphs of the form y = x2 + bx + c | 294 | ||
| Investigation 6: Graphs of the form y = ax2 | 295 | ||
| Investigation 7: Graphs of the form y = a(x – h)2 + k | 296 | ||
| H | Axes intercepts | 298 | |
| Investigation 8: Axes intercepts | 299 | ||
| I | Graphs from axes intercepts | 301 | |
| J | Axis of symmetry and vertex | 305 | |
| Investigation 9: Solving quadratic equations graphically | 307 | ||
| K | Where functions meet | 308 | |
| L | Quadratic modelling | 309 | |
| Graphics calculator investigation: Tunnels and trucks | 313 | ||
| Review set 9A | 314 | ||
| Review set 9B | 314 | ||
| Review set 9C | 315 | ||
| 10 | NUMERICAL TRIGONOMETRY | 317 | |
| A | Right angled triangle trigonometry | 318 | |
| B | The trigonometric ratios | 319 | |
| Investigation 1: Complementary angles | 320 | ||
| Investigation 2: Hipparchus and the universe | 323 | ||
| C | Trigonometric problem solving | 328 | |
| D | Constructing trigonometric formulae | 331 | |
| E | 3-dimensional problem solving | 332 | |
| F | Areas of triangles | 337 | |
| G | The cosine rule | 339 | |
| H | The sine rule | 341 | |
| Investigation 3: The ambiguous case | 342 | ||
| I | Using the sine and cosine rules | 343 | |
| Review set 10A | 349 | ||
| Review set 10B | 351 | ||
| Review set 10C | 352 | ||
| 11 | PERIMETER, AREA AND VOLUME | 355 | |
| A | Conversion of units | 356 | |
| B | Perimeter | 359 | |
| C | Area | 364 | |
| D | Problem solving with areas | 370 | |
| Investigation 1: Rectangles of fixed perimeter | 373 | ||
| E | Surface area | 374 | |
| F | Volume | 379 | |
| G | Capacity | 385 | |
| Investigation 2: Minimising material | 388 | ||
| H | Density | 388 | |
| I | Harder applications | 390 | |
| Review set 11A | 391 | ||
| Review set 11B | 392 | ||
| Review set 11C | 394 | ||
| 12 | SEQUENCES AND SERIES | 395 | |
| A | Number patterns | 396 | |
| Spreadsheet Investigation: Number patterns | 397 | ||
| B | Sequences of numbers | 398 | |
| C | Arithmetic sequences | 401 | |
| D | Geometric sequences | 404 | |
| E | Compound interest | 408 | |
| F | Growth and decay | 410 | |
| G | Series | 411 | |
| Investigation: Van Koch’s snowflake curve | 417 | ||
| Review set 12A | 418 | ||
| Review set 12B | 419 | ||
| Review set 12C | 419 | ||
| 13 | FINANCIAL MATHEMATICS | 421 | |
| A | Foreign exchange | 422 | |
| B | Simple interest | 427 | |
| C | Compound interest | 430 | |
| Investigation 1: Doubling time | 434 | ||
| D | Depreciation | 443 | |
| E | Personal loans | 445 | |
| Investigation 2: Buying a car | 448 | ||
| F | The effect of inflation | 449 | |
| Review set 13A | 451 | ||
| Review set 13B | 452 | ||
| 14 | PROBABILITY | 453 | |
| A | Experimental probability | 455 | |
| Investigation 1: Tossing drawing pins | 455 | ||
| B | Chance investigations | 456 | |
| Investigation 2: Coin tossing experiments | 457 | ||
| Investigation 3: Dice rolling experiments | 458 | ||
| C | Estimating probabilities from data | 460 | |
| D | Sample space | 462 | |
| E | Theoretical probability | 464 | |
| Investigation 4: A probability experiment | 467 | ||
| F | Using grids to find probabilities | 467 | |
| G | Compound events | 469 | |
| Investigation 5: Probabilities of compound events | 469 | ||
| Investigation 6: Revisiting drawing pins | 470 | ||
| H | Using tree diagrams | 473 | |
| I | Sampling with and without replacement | 477 | |
| Investigation 7: Sampling simulation | 480 | ||
| J | Probabilities from Venn diagrams | 481 | |
| K | Laws of probability | 484 | |
| L | Independent events revisited | 489 | |
| Review set 14A | 490 | ||
| Review set 14B | 492 | ||
| Review set 14C | 493 | ||
| 15 | LOGIC | 495 | |
| A | Propositions | 496 | |
| B | Compound propositions | 499 | |
| C | Truth tables and logical equivalence | 503 | |
| D | Truth tables for three propositions | 505 | |
| E | Implication | 506 | |
| F | Converse, inverse and contrapositive | 508 | |
| G | Valid arguments | 510 | |
| Investigation: Syllogisms | 512 | ||
| H | Arguments with three propositions | 513 | |
| Review set 15A | 514 | ||
| Review set 15B | 515 | ||
| Review set 15C | 516 | ||
| 16 | EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS | 517 | |
| A | Evaluating exponential functions | 518 | |
| B | Graphing simple exponential functions | 519 | |
| Investigation 1: Exponential graphs | 521 | ||
| Investigation 2: Solving exponential equations | 523 | ||
| C | Exponential growth | 523 | |
| D | Exponential decay | 526 | |
| E | Periodic functions | 528 | |
| F | Sine functions | 532 | |
| Investigation 3: The family y = Asinx | 533 | ||
| Investigation 4: The family y = sin Bx, B > 0 | 533 | ||
| Investigation 5: The family y = sin x + c | 535 | ||
| G | Cosine functions | 536 | |
| H | Modelling using sine and cosine functions | 537 | |
| I | Equations involving sine and cosine | 539 | |
| J | Using sine and cosine models | 541 | |
| Review set 16A | 543 | ||
| Review set 16B | 543 | ||
| Review set 16C | 544 | ||
| Review set 16D | 544 | ||
| 17 | MORE FUNCTIONS | 545 | |
| A | Cubic polynomials | 547 | |
| Investigation 1: Graphing some families of cubics | 548 | ||
| Investigation 2: Cubics in linear factored form | 550 | ||
| B | Quartic polynomials | 554 | |
| Investigation 3: Graphing quartics | 554 | ||
| C | The rectangular hyperbola | 557 | |
| Investigation 4: The family of curves y = A/x | 558 | ||
| Investigation 5: Functions of the form y = A/(x&ndashh) + k | 559 | ||
| Investigation 6: Higher order rational functions | 563 | ||
| D | Higher order rational functions | 564 | |
| E | Unfamiliar functions | 564 | |
| F | Where functions meet | 566 | |
| Review set 17A | 569 | ||
| Review set 17B | 570 | ||
| 18 | TWO VARIABLE STATISTICS | 571 | |
| A | Correlation | 573 | |
| B | Measuring correlation | 577 | |
| C | Least squares regression | 584 | |
| Investigation: Spearman’s rank order correlation coefficient | 590 | ||
| D | The χ2 test of independence | 591 | |
| Review set 18A | 597 | ||
| Review set 18B | 598 | ||
| 19 | INTRODUCTORY DIFFERENTIAL CALCULUS | 601 | |
| A | Rate of change | 602 | |
| Investigation 1: Instantaneous speed | 606 | ||
| B | Derivatives | 608 | |
| Investigation 2: The derivative of x3 | 610 | ||
| Investigation 3: The derivative of xn | 610 | ||
| C | The idea of a limit | 611 | |
| Investigation 4: The gradient of a tangent | 611 | ||
| D | The derivative function | 612 | |
| E | Simple rules of differentiation | 614 | |
| Investigation 5: Simple rules of differentiation | 614 | ||
| F | Tangents to curves | 617 | |
| G | The second derivative | 619 | |
| H | Changing shape | 620 | |
| I | Stationary points | 624 | |
| J | Rates of change | 627 | |
| K | Optimisation | 630 | |
| Review set 19A | 636 | ||
| Review set 19B | 636 | ||
| Review set 19C | 637 | ||
| ANSWERS | 639 | ||
| INDEX | 697 | ||
