Real-World Problems for Secondary School Mathematics Students: Case Studies

Cover......Page 1
Real-World Problems for Secondary School Mathematics Students......Page 4
ISBN: 9789460915420 (hardback)......Page 5
TABLE OF CONTENTS......Page 6
PREFACE......Page 8
ACKNOWLEDGEMENTS......Page 9
1. MODELLING IN PROBABILITY AND STATISTICS: Key Ideas and Innovative Examples......Page 12
INNOVATIVE EXAMPLES OF PROBABILISTIC MODELLING......Page 13
Blood Samples Modelled with Binomial Probabilities......Page 14
Lifetime of Bulbs Modelled with Normal Distribution......Page 16
Call Times and Cost – the Exponential and Poisson Distributions Combined......Page 19
Spam Mail – Revising Probabilities with Bayes’ Formula......Page 20
THE USUAL APPROACH TOWARDS TEACHING PROBABILITY......Page 22
MODELLING THE NOWITZKI TASK......Page 23
Basic Assumptions of Bernoulli Processes......Page 24
Investigating and Modelling the Unknown Value of p......Page 26
More about Assumptions – A Homogenizing Idea ‘Behind’ the Binomial Distribution......Page 29
The Question ‘Nowitzki Scores at Most Four Times in a Series’......Page 31
KEY IDEA BEHIND VARIOUS DISTRIBUTIONS......Page 33
a. Binomial Distribution – Repeated Independent Trials......Page 34
c. Poisson Distribution – Pure Random Events in Time......Page 35
d. Geometric Distribution – Memory-Less Waiting for an Event......Page 37
e. Exponential Distribution – Memory-Less Waiting for Events in Time......Page 38
g. Normal Distribution – the Hypothesis of Independent Elementary Errors......Page 39
SOLUTIONS TO THE STATISTICAL PART OF THE NOWITZKI TASK......Page 42
Solution to the Second Statistical Part – Nowitzki Weaker Away Than at Home?......Page 43
Validity of Assumptions – Contrasting Probabilistic and Statistical Point of Views......Page 45
Fisher’s exact test.......Page 46
Some Conclusions on the Statistical Modelling of the Nowitzki Task......Page 47
Dealing with the Inherent Assumptions......Page 48
Empirical Research – Generalizing Results from Limited Data......Page 49
CONCLUSIONS......Page 50
REFERENCES......Page 53
INTRODUCTION......Page 56
The Problem of CO2 Emission......Page 57
Usable Solar Energy......Page 58
Photovoltaic Plant and Series Connected Efficiencies......Page 64
Wind Energy Converter......Page 65
Wind Energy Development......Page 67
Betz’ Law and Differentiation......Page 68
Biomass and Reduction of CO2 Emissions......Page 70
Automobile Energy Consumption......Page 71
Automobiles: Forces, Energy and Power......Page 73
OUTLOOK AND FINAL REMARKS......Page 76
REFERENCES......Page 77
INTRODUCTION......Page 78
Modular Arithmetic......Page 79
Check Digit......Page 81
Images as Bits......Page 83
Phase Modulation......Page 85
Coding the Information......Page 87
Geometries......Page 88
Digital Music......Page 92
Physical Structure......Page 93
Error Detection and Correction......Page 94
Art Work......Page 95
REFERENCES......Page 96
Some Background......Page 98
Activity 1: Drawing Ellipses......Page 100
Activity 2: Distance between Mars and the Earth......Page 101
DEALING WITH LARGE NUMBERS: TRAVELLING BETWEEN THE EARTH AND MARS......Page 104
Activity 3: Velocities and Distances......Page 105
CONCLUSION......Page 108
ACKNOWLEDGEMENTS......Page 109
CAN PAPER TALK?......Page 110
2-Dimensional Bar Codes......Page 112
The Structure of the QR-Code......Page 113
MODELLING AND OPTIMISATION: THE QUATTRO-CODE......Page 114
Error Correction......Page 115
Check Sums for 2 x 2 Pixels......Page 117
The Layout of the Quattro Code......Page 119
FURTHER DEVELOPMENTS OF THE QUATTRO CODE......Page 120
Knowledge documentation and presentation skills:......Page 121
REFERENCES/BIBLIOGRAPHY......Page 122
PRACTICE ORIENTATED MATHEMATICS EDUCATION (PRINCIPAL IDEAS)......Page 124
PRACTICE ORIENTATED MATHEMATICS EDUCATION (METHODICAL RUN)......Page 125
Expenses of Buying an Automobile or a Motor Cycle......Page 126
Extending a Loft......Page 127
Dyke Raising......Page 129
Sound Nutrition......Page 130
Architecture in Our City and Aesthetical Aspects in Works of Art Based on Geometry......Page 131
NOTES......Page 132
REFERENCES......Page 133
POLITICAL POLLS......Page 134
Sampling Methods......Page 135
Sample Size......Page 137
Non Response......Page 142
AN ACTIVITY FOR THE CLASSROOM......Page 143
Contacting the Students......Page 144
SUMMARY......Page 145
REFERENCES......Page 146
INTRODUCTION......Page 148
MATHEMATICAL LITERACY AND MODELLING......Page 149
Level 2 – Independent Modelling......Page 150
WATERTANK......Page 151
Mathematical Modelling......Page 152
The Idealized Shot......Page 154
Reconstructing the trajectory of a shot.......Page 155
Covered topics in mathematics.......Page 162
REFERENCES......Page 163
INTRODUCTION......Page 166
Solution:......Page 167
Solution:......Page 168
RADIATION EXPOSURE......Page 169
Solution:......Page 170
AURORA PROBLEMS......Page 171
Solution:......Page 174
Notes for Teachers:......Page 175
Solution:......Page 176
DETERMINING VISIBLE SURFACE AREA......Page 177
CONCLUSION......Page 181
REFERENCES......Page 182
CALCULATING THE ODDS......Page 184
BONUS BALL OR POWERBALL......Page 189
EUROMILLIONS......Page 192
SOME IDEAS FOR USE IN THE MATHEMATICS CLASSROOM......Page 195
REFERENCES......Page 196
INTRODUCTION......Page 198
PHASE 2: Causal Loop Diagrams Using Software (VENSIM PLE3)......Page 199
PHASE 4: Stock and Flow Diagrams......Page 202
Students’ Solutions:......Page 203
DIDACTICS AND FURTHER DETAILS......Page 206
Phase 2: Causal Loop Diagrams......Page 207
A Few Examples:......Page 208
Phase 4: Stock and Flow Diagrams......Page 210
A few examples might be useful:......Page 212
DIDACTICS......Page 217
NOTES......Page 218
REFERENCES/BIBLIOGRAPHY......Page 219
CONVERTING A TRY IN RUGBY......Page 220
Investigation Using Geometry......Page 221
Proof......Page 223
A Different Approach (Using Calculus)......Page 224
ANGLES IN SNOOKER......Page 226
Two Cushion Shots......Page 228
n Cushions Shots......Page 229
Shortest Path (Cue Ball Travels Least Possible Distance)......Page 230
REFERENCES......Page 231
Preliminary Remarks......Page 232
OUTLINE OF THE LESSON......Page 233
Excursion 1: Some Hints About the Mathematical Background......Page 240
Excursion 2: Some Ideas About the Plausibility of the Formula......Page 242
REFERENCES......Page 247
LEARNING MATHEMATICS AND MODELLING BY CALCULTING EGGS SOME ARGUMENTS FROM MATHEMATICS EDUCATION......Page 250
PROBLEMS IN THE REALM OF STUDENTS’ EXPERIENCES IN MATHEMATICS EDUCATION......Page 251
THE EGG MOTIVATION AND STARTING EXAMPLES......Page 252
THE EGG STARTING POINTS FOR CALCULTIONS......Page 253
Definition 1......Page 254
Definition 2......Page 258
CALCULATING EGGS WITH COMPUTERS......Page 261
REFERENCES......Page 266
IMAGES IN INFORMATICS......Page 268
Smoothing in a Spreadsheet......Page 269
Filters in General......Page 273
Pupils’ Experiments (further (linear) filters)......Page 274
What is it, What is it Needed for and How Can it be Done?......Page 275
Example: Differentiation in the x-direction and Catching Inadmissible Values......Page 276
Implementation and its Problems......Page 277
Example to Test the General Spreadsheet......Page 280
Comments......Page 281
REFERENCES......Page 282
INTRODUCTION......Page 284
MODELLING THROUGH THE HELP OF TECHNOLOGY......Page 286
First step – Constructing a suitable model for the given situation.......Page 287
Fourth step – Solving the function.......Page 288
Fifth step – Interpreting and arguing the solution.......Page 289
Advancements in the model.......Page 290
REFERENCES......Page 291
LIST OF CONTRIBUTORS......Page 292