Learning Mathematics With ClassWiz

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(2) PREFACE. Over 40 years, the scientific calculator has evolved from being a computational device for scientists and engineers to becoming an important educational tool. What began as an instrument to answer numerical questions has evolved to become an affordable, powerful and flexible environment for students and their teachers to explore a wide range of mathematical ideas and relationships. The significant calculator developments of recent years, together with advice from experienced teachers, have culminated in the advanced scientific calculators, the CASIO fx-991 EX and the CASIO fx-570 EX, with substantial mathematical capabilities, including a spreadsheet, and extensive use of natural displays of mathematical notation. This publication comprises a series of modules to help make best use of the opportunities for mathematics education afforded by these developments. The focus of the modules is on the use of the ClassWiz in the development of students’ understanding of mathematical concepts and relationships, as an integral part of the development of mathematical meaning for the students. While meeting the computational needs of students throughout secondary school, and beyond, the ClassWiz can also be used to advantage by students to support their initial learning of the mathematical ideas involved; the calculator is not only a device to be used to undertake or to check computations, after the mathematics has been understood. The mathematics involved in the modules spans a wide range from the early years of secondary school to the early undergraduate years, from early ideas of number and algebra through to the study of calculus, probability and statistics, as well as advanced topics such as vectors, matrices and complex numbers. I expect that readers will decide which modules suit their purposes. Although mathematics curricula vary across different countries, I am confident that the mathematical ideas included in the modules will be of interest to mathematics teachers and their students across international boundaries. The text of each modules is intended to be read by both teachers and students, to understand how the ClassWiz is related to various aspects of mathematics, and also to help them to use it efficiently. Each module contains a set of Exercises, focusing on calculator skills relevant to the mathematics associated with the module. In addition, a set of exploratory Activities is provided for each module after the first, to illustrate some of the ways in which the calculator can be used to explore mathematical ideas through the use of the calculator; these are not intended to be exhaustive, and I expect that teachers will develop further activities of these kinds to suit their students. The Notes for Teachers in each module provide answers to exercises, as well as some advice about the classroom use of the activities (including answers where appropriate). Permission is given for the reproduction of any of the materials for educational purposes. The material in many of these modules draws significantly on materials developed earlier by Marian Kemp and myself, refining and extending those materials to take advantage of the many innovative features included in the ClassWiz. I am grateful to CASIO for supporting the development of these materials, and appreciate in particular the assistance of Mr Yoshino throughout the developmental process, as well as the careful proof-reading of Mr Rabieh Al Halabi. I hope that users of these materials enjoy working with the calculator as much as I have enjoyed developing the materials and I wish both teachers and their students a productive engagement with mathematics through the use of the ClassWiz. Barry Kissane Murdoch University, Western Australia.

(3) Table of Contents Module 1: Introduction to ClassWiz ········································································1 Entering and editing commands ········································································1 Mathematical commands ················································································2 Recalling commands ·····················································································4 Scientific and engineering notation ····································································5 Calculator modes ·························································································7 OPTN commands ·························································································8 SET UP ····································································································8 Memories ································································································ 12 Initializing the calculator ·············································································· 13 Exercises and Notes for teachers ····································································· 14 Module 2: Representing numbers ········································································· 16 Representing decimals ················································································· 16 Representing fractions ················································································· 17 Representing percentages ·············································································· 18 Recurring decimals ····················································································· 18 Powers···································································································· 20 Factors···································································································· 22 Scientific notation ······················································································ 23 Roots ····································································································· 24 Reciprocals ······························································································ 25 Rational and irrational numbers ······································································ 26 Exercises, Activities and Notes for teachers ························································ 28 Module 3: Functions ························································································· 31 Evaluating expressions and functions ································································ 31 Comparing expressions ················································································ 32 Using tables of values ·················································································· 32 Linear and quadratic functions ········································································ 34 Cubic functions ························································································· 36 Reciprocal functions···················································································· 37 Maximum and minimum values ······································································ 39 Intersection of two graphs ············································································· 40 Exercises, Activities and Notes for teachers ························································ 42. i.

(4) Module 4: Equations and inequalities ···································································· 45 Equations, inequalities and a table ··································································· 45 Using two tables ························································································ 47 Automatic equation solving ··········································································· 48 Automatic inequality solving·········································································· 49 Solving quadratic equations and inequalities ······················································· 50 Systems of linear equations ··········································································· 51 Using the Solver ························································································ 52 Ratio and proportion ··················································································· 55 Exercises, Activities and Notes for teachers ························································ 57 Module 5: Trigonometry ···················································································· 60 Trigonometry and right triangles ····································································· 60 Tables of values ························································································· 62 Exact values ····························································································· 63 Radian measure ························································································· 64 Gradian measure ························································································ 65 Solving triangles with the Sine Rule ································································· 66 Solving triangles with the Cosine Rule ······························································ 67 The Pythagorean Identity ·············································································· 68 Coordinate systems ····················································································· 69 Trigonometric equations ··············································································· 70 Exercises, Activities and Notes for teachers ························································ 71 Module 6: Exponential and logarithmic functions ····················································· 74 Exponents and roots ···················································································· 74 Exploring exponential functions ······································································ 75 Using exponential models ············································································· 77 Logarithms······························································································· 78 Properties of logarithms ··············································································· 79 Logarithms to other bases ············································································· 80 e and natural logarithms ··············································································· 82 Exercises, Activities and Notes for teachers ························································ 85. ii.

(5) Module 7: Matrices ·························································································· 88 Defining matrices ······················································································· 88 Matrix arithmetic ······················································································· 89 Matrix inversion ························································································ 91 Transformation matrices ··············································································· 93 Matrices and equations ················································································· 94 Exercises, Activities and Notes for teachers ························································ 95 Module 8: Vectors ···························································································· 98 Representing vectors ··················································································· 98 Vector magnitude and direction ······································································ 99 Vector arithmetic ······················································································101 An example from sailing··············································································103 Dot product ·····························································································104 Three-dimensional vectors ···········································································105 Cross product ···························································································106 Exercises, Activities and Notes for teachers ·······················································109 Module 9: Further numbers···············································································112 Scientific constants ····················································································112 Measurement conversions ············································································113 Numbers to other bases ···············································································114 Binary logical operations ·············································································116 Complex numbers ·····················································································117 Argand diagrams ·······················································································118 Polar form ······························································································119 Powers and roots of complex numbers ·····························································120 Exercises, Activities and Notes for teachers ·······················································122 Module 10: Univariate statistics ··········································································125 Getting started with statistics·········································································125 Entering, editing and checking data ·································································125 Retrieving statistics ····················································································127 Determining normal probabilities ···································································129 Frequency data ·························································································130 Inferential statistics ····················································································132 Exercises, Activities and Notes for teachers ·······················································134. iii.

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