Table Of ContentColourful mathematics
10
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Colourful mathematics
József Kosztolányi
István Kovács
Klára Pintér
János Urbán
István Vincze
M he ati s
textbook 10
Mozaik Education - Szeged, 2015
Authors: JÓZSef Kosztolányi PhD ■ associate professor
István Kovács ■ secondary grammar school teacher
Klára Pintér ■ college senior lecturer
János Urbán PhD ■ secondary grammar school teacher
István Vincze • secondary grammar school teacher
Consultants: József Németh PhD • associate professor
József né Szigeti • secondary school teacher
Translated by: Orsolya Koros-Rónai
Editor: Katalin Tóth
Typography: Tamás Reményfy
illustrated by: István Ábrahám
Photos: Mozaik Archívum - Margit Hilbert Phd;
International photo agencies
All rights reserved, including copying, the publishing of shortened
or lengthened versions of the book. The whole book or any part thereof
cannot be copied in any form (microfilm, photocopy or other carrier)
withouth the written consent of the publishing company.
isbn 978 963 697 772 6
Copyright: Mozaik Education - Szeged, 2015
Mathematical reasoning
1. What does it imply? ........................................................... 10
2. The pigeonhole principle ..................................................... 21
3. Arrangement (ordering) problems ....................................... 29
4. Picking problems ............................................................... 32
Computing the root
1. Rational numbers, irrational numbers .................................. 36
2. The identities (laws) of the square root ................................ 40
3. Applying the identities (laws) of the square root .................... 44
4. The nth root of numbers ..................................................... 50
5. The identities (laws) of the nth root ...................................... 53
The quadratic equation
1. The quadratic equation and function .................................... 60
2. The quadratic formula ........................................................ 64
3. The zero product form. The relation between the roots
and the coefficients ............................................................ 69
4. Equations of higher degree which can be reduced
to quadratic equations ........................................................ 74
5. Quadratic inequalities ......................................................... 80
6. Parametric quadratic equations (higher level courseware) ...... 84
7. Equations involving square roots ......................................... 90
8. Quadratic simultaneous equations ....................................... 96
9. Arithmetic and geometric mean .......................................... 101
10. Extreme value exercises (higher level courseware) ............... 106
11. Problems leading to quadratic equations .............................. 110
Geometry
Widening the knowledge about circles
1. Reminder........................................................................... 116
2. The theorem of the central
and inscribed/tangent-chord angles .................................... 117
3. The theorem of inscribed angles; the arc of viewing angles ... 121
4. The theorem of inscribed quadrilaterals
(higher level courseware) ................................................... 125
The similarity transformation and its applications
1. Parallel intercepting lines, parallel intercepting line segments
(higher level courseware) .................................................. 129
6
2. The angle bisector theorem (higher level courseware) ........... 135
3. The transformation of central dilation (or homothety) ............ 137
4. The similarity transformation ............................................... 141
5. Similarity of figures; the simple cases of similar triangles ...... 143
6. A few applications of similarity ............................................ 147
7. The ratio of the area of similar planar figures ........................ 154
8. The ratio of the volume of similar solids .............................. 158
Trigonometric functions of acute angles
1. Determining distances with the help of similarity .................. 161
2. Trigonometric functions of acute angles ............................... 164
3. Relations between the trigonometric functions
of acute angles .................................................................. 168
4. Trigonometric functions of special angles 172
5. Determining several data of a triangle with the help
of trigonometric functions .................................................. 175
6. Calculations in the plane and in space with the help
of trigonometric functions .................................................. 180
Vectors
1. The concept of a vector; the sum and the difference of vectors;
scalar multiplication of vectors (reminder) ........................... 184
2. Expressing vectors as the sum of components
in different directions .......................................................... 188
3. Applying vectors in the plane and in space ........................... 194
4. Vectors in the coordinate system, the coordinates of a vector,
operations with vectors given with coordinates .................... 199
Trigonometric functions
1. The definition and the simple properties of the sine
and the cosine function ...................................................... 204
2. The graph of the sine function ............................................. 209
3. The graph of the cosine function, equations, inequalities ......... 214
4. The tangent and the cotangent function ............................... 221
5. Compound exercises and applications ................................. 228
6. Geometric applications ....................................................... 232
Calculation of probability
1. Events............................................................................... 238
2. Operations with events....................................................... 243
3. Experiments, frequency, relative frequency, probability ........... 248
4. The classical model of probability ....................................... 251
7
Guide to use the course book
The notations and highlights used in the book help with acquiring the courseware.
- The train of thought of the worked examples show samples how to understand
the methods and processes and how to solve the subsequent exercises.
- The most important definitions and theorems are denoted by colourful highlights.
- The parts of the courseware in small print and the worked examples noted
in claret colour help with deeper understanding of the courseware. These pieces
of knowledge are necessary for the higher level of graduation.
- Figures, the key points of the given lesson, review and explanatory parts along
with interesting facts of the history of mathematics can be found on the margin.
The difficulty level of the examples and the appointed exercises is denoted by three
different colours:
Yellow: drilling exercises with basic level difficulty; the solution and drilling
of these exercises is essential for the progress.
Blue: exercises the difficulty of which corresponds to the intermediate level
of graduation.
Claret: problems and exercises that help with preparing for the higher level
of graduation.
These colour codes correspond to the notations used in the Colourful mathematics
workbooks of Mozaik Education. The workbook series contains more than
3000 exercises which are suitable for drilling, working on in lessons and which
help with preparing for the graduation.
The end results of the appointed exercises can be found on the following
website: www.mozaik.info.hu. Website www.mozaweb.com offers more help
material for processing with the course book.
♦ ♦ ♦
Mathematics, “ratio” and logic way of thinking are probably the most efficient
tools of cognition of our world, which sometimes associate with unexplainable
phenomena. These are inseparable from Homo sapiens and these make
the everyday activities complete.
A few thoughts from those who have experienced all this:
"So what is ratio the human intellect created logic out of? It is obvious that it is ‘there in ’
the nature, otherwise it would not be possible to understand the nature with the help
of rational tools. Ratio unites humans, animals and nature. ” (Imre Kertész, Nobel prlze-
winning Hungarian writer)
“Through the centuries the collective awareness of mathematicians created its own
universe. Where it is I do not know - and I think the word "where" also loses its meaning
here but I can assure the reader: this mathematical universe is all too real to those
who live in it. The mankind could pierce into the mystery of the surrounding world the
most deeply right by means of Mathematics. ” (Ian Stewart)
“The strict proof is usually the last step! Before that many conjectures are needed, and
for these aesthetic belief is extremely important. ” (Roger Penrose)
The Authors wish productive work and learning.
René Descartes (1569-1650) was looking for a general method to solve problems
in his work "Regulae ad directionem ingenii’’ (Rules for the Direction of the Mind).
His idea was firstly to express every problem as a mathematical problem,
then secondly to express every mathematical problem as an algebraic one,
and then finally to solve these in the form of equations.
For the development of our way of thinking it is also necessary
to observe the structure, the schemes of the reflective action.
Henceforth we are going to apply a few methods
which give a useful strategy for solving
many problems. The more strategies we know,
the greater our chance to be able to solve
the problems facing us.
