| introduction |
|
xiii | |
|
|
|
1 | (13) |
|
|
|
2 | (1) |
|
|
|
2 | (2) |
|
transformation of a formula |
|
|
4 | (1) |
|
examples of using formulae |
|
|
5 | (1) |
|
an illustration from numbers |
|
|
6 | (1) |
|
|
|
7 | (1) |
|
examples of generalizing patterns |
|
|
8 | (2) |
|
letters represent numbers, not quantities |
|
|
10 | (1) |
|
examples of algebraic forms |
|
|
11 | (3) |
|
elementary operations in algebra |
|
|
14 | (18) |
|
|
|
15 | (1) |
|
|
|
15 | (1) |
|
algebraic expression, terms |
|
|
16 | (1) |
|
|
|
16 | (1) |
|
|
|
17 | (1) |
|
addition and subtraction of like terms |
|
|
17 | (1) |
|
|
|
18 | (1) |
|
|
|
18 | (1) |
|
evaluation by substitution |
|
|
19 | (2) |
|
|
|
21 | (1) |
|
|
|
22 | (1) |
|
multiplication of powers of a number |
|
|
23 | (1) |
|
|
|
24 | (1) |
|
|
|
25 | (2) |
|
|
|
27 | (1) |
|
|
|
27 | (2) |
|
multiplication and division |
|
|
29 | (3) |
|
brackets and operations with them |
|
|
32 | (9) |
|
|
|
33 | (1) |
|
addition and subtraction of expressions within brackets |
|
|
34 | (2) |
|
|
|
36 | (2) |
|
|
|
38 | (1) |
|
|
|
39 | (2) |
|
positive and negative numbers |
|
|
41 | (12) |
|
the scale of a thermometer |
|
|
42 | (1) |
|
|
|
42 | (1) |
|
a commercial illustration |
|
|
42 | (1) |
|
motion in opposite directions |
|
|
43 | (1) |
|
positive and negative numbers |
|
|
44 | (1) |
|
|
|
45 | (1) |
|
graphical representation of the number line |
|
|
45 | (1) |
|
operations with negative numbers |
|
|
46 | (1) |
|
addition of positive and negative numbers |
|
|
46 | (1) |
|
|
|
47 | (1) |
|
|
|
47 | (2) |
|
|
|
49 | (1) |
|
|
|
50 | (1) |
|
summary of rules of signs for multiplication and division |
|
|
50 | (1) |
|
powers, squares and square roots |
|
|
51 | (2) |
|
expressions and equations |
|
|
53 | (13) |
|
understanding expressions |
|
|
54 | (1) |
|
|
|
55 | (4) |
|
|
|
59 | (2) |
|
|
|
61 | (2) |
|
an introduction to solving equations |
|
|
63 | (3) |
|
|
|
66 | (11) |
|
|
|
67 | (1) |
|
|
|
67 | (2) |
|
|
|
69 | (4) |
|
problems leading to simple equations |
|
|
73 | (4) |
|
|
|
77 | (10) |
|
practical importance of formulae |
|
|
78 | (1) |
|
|
|
78 | (1) |
|
|
|
78 | (2) |
|
transformation of formulae |
|
|
80 | (1) |
|
|
|
81 | (3) |
|
|
|
84 | (1) |
|
|
|
85 | (2) |
|
|
|
87 | (12) |
|
simple equations with two unknown quantities |
|
|
88 | (1) |
|
solution of simultaneous equations |
|
|
88 | (1) |
|
|
|
89 | (5) |
|
problems leading to simultaneous equations |
|
|
94 | (1) |
|
|
|
94 | (5) |
|
|
|
99 | (10) |
|
the idea of an inequality |
|
|
100 | (1) |
|
representing inequalities |
|
|
100 | (3) |
|
|
|
103 | (3) |
|
|
|
106 | (1) |
|
simultaneous inequalities |
|
|
106 | (3) |
|
graphical representation of quantities |
|
|
109 | (12) |
|
the object of graphical work |
|
|
110 | (1) |
|
|
|
110 | (1) |
|
|
|
111 | (1) |
|
|
|
112 | (1) |
|
examples of graphs and their uses |
|
|
113 | (2) |
|
an example from electricity |
|
|
115 | (1) |
|
an example from mechanics |
|
|
116 | (5) |
|
straight-line graphs; coordinates |
|
|
121 | (18) |
|
|
|
122 | (1) |
|
the law represented by a straight-line graph |
|
|
123 | (2) |
|
graph of an equation of the first degree |
|
|
125 | (1) |
|
|
|
126 | (2) |
|
position in a plane; coordinates |
|
|
128 | (3) |
|
a straight line as a locus |
|
|
131 | (2) |
|
equation of any straight line passing through the origin |
|
|
133 | (1) |
|
graphs of straight lines not passing through the origin |
|
|
134 | (2) |
|
graphical solution of simultaneous equations |
|
|
136 | (3) |
|
using inequalities to define regions |
|
|
139 | (11) |
|
|
|
140 | (1) |
|
regions above and below straight lines |
|
|
140 | (4) |
|
greatest or least values in a region |
|
|
144 | (1) |
|
|
|
145 | (5) |
|
multiplying algebraical expressions |
|
|
150 | (11) |
|
when one factor consists of one term |
|
|
151 | (1) |
|
product of expressions with two terms |
|
|
151 | (2) |
|
when the coefficients of the first terms are not unity |
|
|
153 | (1) |
|
multiplication of an expression with three terms |
|
|
154 | (2) |
|
square of an expression with two terms |
|
|
156 | (1) |
|
square of an expression with three terms |
|
|
157 | (1) |
|
cube of an expression with two terms |
|
|
157 | (2) |
|
product of sum and difference |
|
|
159 | (2) |
|
|
|
161 | (14) |
|
the process of finding factors |
|
|
162 | (1) |
|
factors consisting of one term only |
|
|
162 | (1) |
|
|
|
162 | (1) |
|
|
|
163 | (1) |
|
|
|
164 | (2) |
|
|
|
166 | (1) |
|
|
|
166 | (2) |
|
|
|
168 | (2) |
|
expressions which are squares |
|
|
170 | (1) |
|
difference of two squares |
|
|
171 | (1) |
|
|
|
171 | (1) |
|
|
|
172 | (2) |
|
sum and difference of two cubes |
|
|
174 | (1) |
|
|
|
174 | (1) |
|
|
|
175 | (9) |
|
|
|
176 | (1) |
|
|
|
176 | (1) |
|
|
|
176 | (2) |
|
multiplication and division |
|
|
178 | (1) |
|
|
|
179 | (3) |
|
simple equations involving algebraical fractions |
|
|
182 | (2) |
|
graphs of quadratic functions |
|
|
184 | (20) |
|
|
|
185 | (1) |
|
dependent and independent variables |
|
|
186 | (1) |
|
|
|
186 | (1) |
|
|
|
187 | (1) |
|
graph of a function of second degree |
|
|
188 | (1) |
|
some properties of the graph of y = x2 |
|
|
189 | (1) |
|
|
|
190 | (1) |
|
|
|
191 | (1) |
|
the graphs of y = x2 ± a, where a is any number |
|
|
192 | (1) |
|
|
|
193 | (1) |
|
graph of y = (x - 1)2 - 4 |
|
|
194 | (1) |
|
the graph y = x2 - 2x - 3 |
|
|
195 | (1) |
|
solution of the equation x2 - 2x - 3 = 0 from the graph |
|
|
196 | (1) |
|
graph of y = 2x2 - 3x - 5 |
|
|
196 | (1) |
|
|
|
197 | (1) |
|
using graphics calculators |
|
|
198 | (2) |
|
using graphs to solve quadratic inequalities |
|
|
200 | (2) |
|
using quadratic inequalities to describe regions |
|
|
202 | (2) |
|
|
|
204 | (20) |
|
|
|
205 | (1) |
|
the method of solution of any quadratic |
|
|
206 | (1) |
|
solution of 2x2 + 5x - 3 = 0 |
|
|
207 | (1) |
|
|
|
207 | (3) |
|
solution of quadratic equations by factorization |
|
|
210 | (1) |
|
|
|
211 | (1) |
|
general formula for the solution of a quadratic equation |
|
|
212 | (1) |
|
solution of the quadratic equation ax2 + bx + c = 0 |
|
|
213 | (1) |
|
|
|
214 | (2) |
|
problems leading to quadratics |
|
|
216 | (3) |
|
simultaneous equations of the second degree |
|
|
219 | (1) |
|
when one of the equations is of the first degree |
|
|
219 | (2) |
|
solving quadratic inequalities |
|
|
221 | (3) |
|
|
|
224 | (12) |
|
|
|
225 | (1) |
|
|
|
225 | (3) |
|
extension of the meaning of an index |
|
|
228 | (1) |
|
|
|
228 | (2) |
|
algebraical consideration of the extension of the meaning of indices |
|
|
230 | (1) |
|
|
|
230 | (1) |
|
|
|
231 | (1) |
|
|
|
231 | (2) |
|
standard forms of numbers |
|
|
233 | (1) |
|
operations with standard forms |
|
|
234 | (2) |
|
|
|
236 | (9) |
|
|
|
237 | (2) |
|
|
|
239 | (1) |
|
rules for the use of logarithms |
|
|
240 | (2) |
|
change of base of a system of logarithms |
|
|
242 | (3) |
|
|
|
245 | (8) |
|
|
|
246 | (1) |
|
|
|
246 | (1) |
|
|
|
247 | (1) |
|
theorems on ratio and proportion |
|
|
247 | (1) |
|
an illustration from geometry |
|
|
248 | (1) |
|
|
|
249 | (1) |
|
|
|
250 | (3) |
|
|
|
253 | (21) |
|
|
|
254 | (1) |
|
examples of direct variation |
|
|
255 | (1) |
|
the constant of variation |
|
|
255 | (1) |
|
|
|
256 | (1) |
|
to find the law connecting two variables |
|
|
256 | (1) |
|
|
|
257 | (1) |
|
y partly constant and partly varying as x |
|
|
258 | (1) |
|
|
|
259 | (2) |
|
y varies as the square of x - that is, y ∞ x2 |
|
|
261 | (1) |
|
y varies as the cube of x - that is, y ∞ x3 |
|
|
262 | (1) |
|
y varies as √x or x1/2, that is, y ∞ √x |
|
|
263 | (1) |
|
inverse variation y ∞ 1/x |
|
|
264 | (1) |
|
|
|
265 | (2) |
|
other forms of inverse variation |
|
|
267 | (1) |
|
|
|
267 | (2) |
|
functions of more than one variable |
|
|
269 | (2) |
|
|
|
271 | (1) |
|
|
|
271 | (3) |
|
the determination of laws |
|
|
274 | (8) |
|
laws which are not linear |
|
|
275 | (1) |
|
y = axn + b. plotting against a power of a number |
|
|
275 | (1) |
|
|
|
276 | (1) |
|
y = axn. use of logarithms |
|
|
277 | (1) |
|
|
|
278 | (4) |
|
rational and irrational numbers, surds |
|
|
282 | (7) |
|
rational and irrational numbers |
|
|
283 | (1) |
|
irrational numbers and the number line |
|
|
284 | (1) |
|
geometrical representation of surds |
|
|
284 | (1) |
|
|
|
285 | (4) |
|
arithmetical and geometrical sequences |
|
|
289 | (24) |
|
|
|
290 | (1) |
|
the formation of a sequence |
|
|
290 | (1) |
|
arithmetic sequences, or arithmetic progressions |
|
|
291 | (1) |
|
any term in an arithmetic sequence |
|
|
291 | (1) |
|
the sum of any number of terms of an arithmetic sequence |
|
|
292 | (1) |
|
|
|
293 | (1) |
|
|
|
293 | (2) |
|
harmonic sequences, or harmonic progressions |
|
|
295 | (1) |
|
geometric sequences or geometric progressions |
|
|
296 | (1) |
|
connection between a geometric sequence and an arithmetic sequence |
|
|
297 | (1) |
|
general term of a geometric sequence |
|
|
297 | (1) |
|
|
|
298 | (2) |
|
the sum of n terms of a geometric sequence |
|
|
300 | (1) |
|
|
|
300 | (1) |
|
increasing geometric sequences |
|
|
301 | (1) |
|
decreasing geometric sequences |
|
|
302 | (1) |
|
|
|
302 | (2) |
|
a geometrical illustration |
|
|
304 | (1) |
|
|
|
305 | (1) |
|
|
|
306 | (2) |
|
simple and compound interest |
|
|
308 | (2) |
|
accumulated value of periodical payments |
|
|
310 | (1) |
|
|
|
310 | (3) |
|
|
|
313 | (8) |
|
permutations and combinations |
|
|
313 | (3) |
|
|
|
316 | (3) |
|
the roots of a quadratic equation |
|
|
319 | (2) |
| Answers |
|
321 | (30) |
| Index |
|
351 | |
Other versions by this Author