Cambridge IGCSE Mathematics Core and Extended - Coursebook - Hodder

9 M athematical investigations
and ICT

Heights and percentiles

The graphs below show the height charts for males and females from
the age of 2 to 20 years in the United States.

CDC Growth Charts: United States

2c0m0 in in
78 78
195 76
190 Stature-for-age percentiles: 76
74 Boys, 2 to 20 years
185 97th
72
180 70 95th 74

90th

75th 72

175 50th 70

170 68 25th 68

165 66 10th 66
5th
3rd 64
160 64
155 62 62

150 60 60

145 58 58

140 56 56

135 54 54

130 52 52

125 50 50

120 48 48

115 46 46

110 44 44

105 42 42

100 40 40

95 38 38

90 36 36

85 34 34

80 32 32

c7m5 30 30
in in

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Age (years)

Note: Heights have been given in both centimetres and inches.

541

9 Mathematical investigations and ICT

CDC Growth Charts: United States

2c0m0 in in
78 78
195 76
190 74 Stature-for-age percentiles: 76
Girls, 2 to 20 years
185 74

72 72
180 70
175 68 97th 70
170 95th 68
165 66 90th 66
75th

160 64 50th 64

155 62 25th 62
10th
150 60 5th 60
3rd

145 58 58

140 56 56

135 54 54

130 52 52

125 50 50

120 48 48

115 46 46

110 44 44

105 42 42

100 40 40

95 38 38

90 36 36

85 34 34

80 32 32

c7m5 30 30
in in

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Age (years)

1 From the graph, find the height corresponding to the 75th percentile
for 16-year-old girls.

2 Find the height which 75% of 16-year-old boys are taller than.
3 What is the median height for 12-year-old girls?
4 Measure the heights of students in your class. By carrying out

appropriate statistical calculations, write a report comparing your
data to that shown in the graphs.
5 Would all cultures use the same height charts? Explain your answer.

Reading ages

Depending on their target audience, newspapers, magazines and books
have different levels of readability. Some are easy to read and others
more difficult.

1 Decide on some factors that you think would affect the readability of
a text.

542

ICT activity

2 Write down the names of two newspapers which you think would
have different reading ages. Give reasons for your answer.

There are established formulae for calculating the reading age of
different texts.

One of these is the Gunning Fog Index. It calculates the reading age

as follows: 2 A 100L
5 n A
age( )Reading= + where

A = number of words

n = number of sentences

L = number of words with 3 or more syllables

3 Choose one article from each of the two newspapers you chose in
Question 2. Use the Gunning Fog Index to calculate the reading ages
for the articles. Do the results support your predictions?

4 Write down some factors which you think may affect the reliability of
your results.

ICT activity

In this activity you will be collecting the height data of all the students
in your class and plotting a cumulative frequency diagram of the results.

1 Measure the heights of all the students in your class.
2 Group your data appropriately.
3 Enter your data into graphing software such as Excel or Autograph.
4 Produce a cumulative frequency diagram of the results.
5 From your graph, find:

a the median height of the students in your class,
b the interquartile range of the heights.
6 Compare the cumulative frequency diagram from your class with one
produced from data collected from another class in a different year
group. Comment on any differences or similarities between the two.

543

Index

Index Numbers of elevation and depression Bhascara 3
405–6 bonuses 74
24-hour clock 83 box-and-whisker plots 536–7
of irregular polygons 329 brackets 26–7, 34
A between a line and a plane
expanding a pair of
acceleration 178, 185–6 429–31 brackets 103, 106–7
accuracy in opposite segments of a
expanding a single
approximation 13–14 circle 333 bracket 102
estimating answers 15 within parallel lines 319
upper and lower at a point 318 Brahmagupta 3
in a quadrilateral 323–4 Buffon’s needle experiment 500
bounds 17–19 in the same segment of a
acute-angled triangles 283 C
acute angles 282 circle 332
addition in a semicircle 327 calculators
sum of exterior angles of a order of operations 26–7
of fractions 36 powers and roots 10–11
of vectors 442 polygon 326
adjacent side, right-angled sum of interior angles of a calculus 219
see also differentiation
triangles 394 polygon 325
algebra supplementary 333 Cardano, Girolamo 439
between a tangent and a centre of enlargement 460
constructing equations centre of rotation 455
127–8, 135 radius 328 chaos theory 475
in a triangle 321 chequered boards
expanding a pair of brackets angle–side–angle (ASA) rule,
103, 106–7 investigation 245
congruent triangles 304 chords 285
expanding a single bracket Apollonius 281
102 approximation 13–14 equal 313
Archimedes 281 circles 285
factorising 104, 108–11 arc of a circle 285
inequalities 142–3 alternate segment
linear equations 126–7 length of 365–6 theorem 333–4
quadratic equations 137–40 area
rearranging formulae 105–6, angle at the centre of 331
of a circle 356–7 angle between a tangent and a
112–13 of a parallelogram 354
simultaneous equations of a rectangle 351 radius 328
of a sector 368 angle in a semicircle 327
130–3, 141 of a trapezium 354, 355 angle in opposite
substitution 105 of a triangle 352, 423
algebraic fractions see also surface area segments 333
addition and subtraction area factors 293, 297 angles in the same
arithmetic sequences 153–4
115–17 Aryabhata 3 segment 332
simplifying 114–15, 118 asymptotes 212–13 arc length 365–6
algebraic indices 122–5 averages 504–5 area 356–7
al-Karaji 101 average speed 173 area of a sector 368
al-Khwarizmi 101 circumference 356–7
Al-Kindi 503 B equal chords 313
alternate angles 319 ICT activity 343
alternate segment back bearings 392–3 tangents from an external
back-to-back stem-and-leaf
theorem 333–4 point 315
angle–angle–side (AAS) rule, diagrams 512 circumference of a circle 285,
bar charts 509, 511
congruent triangles 304 base of a triangle 352 356–7
angles 282 bearings 392–3 class intervals, histograms 527–8
Bernoulli family 391 column vectors 440–1
alternate segment combined events 488
theorem 333–4
tree diagrams 488–9,
at the centre of a circle 331 491, 494

544

common difference, histograms 525–8 E Index
sequences 153 pictograms 510
pie charts 513–15 earnings 73–4
common ratio, sequences 160 scatter diagrams 519–20 Egyptian mathematics 345
complement of a set 87–8 stem-and-leaf elements of a set 85
completing the square 140 elevation, angles of 405–6
composite functions 243 diagrams 511–12 empty set (∅) 86
compound interest 78–80, tally charts 510 enlargements 460–1
data interpretation
161–2 positive correlation 519–20 negative 463
compound measures 56–8 types of correlation 521 equation of a straight line 257–8
conditional probability 492–4 deceleration 178, 185–6
cones decimal places 13, 14 drawing straight-line
decimals 31–2 graphs 266
construction activity 388 conversion from
surface area 382 in form ax + by + c = 0 263
volume 377–9 fractions 38 general equation 262–3
congruent shapes 301 conversion to and from graphical solution of
triangles 284, 303–5
constant of proportionality 166 percentages 44 simultaneous
continuous data 513 conversion to fractions 39 equations 267–8
conversion graphs 172 fraction and percentage parallel lines 265
converting from one unit to through two points 270
equivalents 43 equations
another 347–8 recurring 39–40 construction of 127–8, 135
area and volume denominator 30 exponential 64–5, 246
density 57 linear 126–7
conversions 349 depreciation 76 quadratic 137–40
coordinates 250 depression, angles of 405–6 simultaneous 130–3, 141,
correlation 519–20 derivatives 224 267–8
of constants 224–5 solution by graphical
types of 521 see also differentiation methods 203–4
corresponding angles 319 Descartes, René 249 trigonometric 412
cosine curve 410 diameter 285 equilateral triangles 283
cosine (cos) ratio 399–400 dice sum investigation 499 equivalent fractions 32, 35
cosine rule 420 difference of two squares 109 estimating answers 15
cost price 75 differentiation 224–7 Euclid 281
cube numbers 6, 163 equation of the tangent at a Euler, Leonhard 391
cube roots 10, 11 exponential equations 64–5
cubic curves 199 given point 233 ICT activity 246
cubic functions, graph gradient of a curve at a exponential functions 201
graph sketching 213–14
sketching 209–11 point 229–30 exponential growth 80
cuboids second derivatives 228–9 exponential sequences
stationary points 234–6 (geometric sequences) 160
surface area 359–60 value of x when the gradient applications 161–2
trigonometry in three
is given 231–2 F
dimensions 426–7 directed numbers 11
cumulative frequency 532–3 direct proportion 50–1, 166–8 factorising 104
discounts 75 difference of two squares 109
quartiles and interquartile discrete data 511, 513 evaluation of
range 534–6 distance 173 expressions 110
by grouping 108
currency conversions 73 shortest distance from a line quadratic expressions 110–11
cyclic quadrilaterals 333 to a point 423
cylinders, surface area 359–60 factors 6–7
from speed–time graphs favourable outcomes 476
D 180–1 Fermat, Pierre de 249, 475
Fibonacci (Leonardo Pisano)
data displays distance–time graphs 174, 176
bar charts 511 non-linear 184, 186–7 439
box-and-whisker plots
536–7 division 33–4
frequency tables 505, 510 of fractions 37
grouped frequency
tables 512–13 double time 74

545

Index football leagues value of x when the gradient is exponential equations 64–5
investigation 98 given 231–2 fractional 68–9
laws of 62, 64
fountain borders graphs negative 64
investigation 341 conversion graphs 172 positive 62
direct proportion 166–8 zero 63
fractional indices 68–9, 123–4 exponential functions 201 see also standard form
fractions 30–1 gradients of curves 201–2 inequalities 24–5
of inequalities 147–9 graphing 147–9
addition and subtraction 36 linear programming 150 ICT activity 246, 279
algebraic 114–18 quadratic functions 193–7 linear 142–3
conversion from decimals 39 reciprocal functions 198 linear programming 150
conversion from recognising and sketching symbols 147
functions 206–14 inflexion, points of 235
percentages 44 solving equations 203–4 integers 4
conversion from recurring travel graphs 174–87, 176 interest
types of 199 compound 78–80, 161–2
decimals 40–1 simple 76–7
conversion to decimals 32, 38 Greek mathematicians 281 interquartile range 534–6
conversion to percentages gross earnings 73–4 intersection of sets (∩) 88
grouped data, mean of 507 inverse functions 241–2
32, 44 grouped frequency tables 512–13 inverse proportion 53–4, 168
decimal and percentage investigations 96
H chequered boards 245
equivalents 43 dice sum 499
equivalent 35 height charts 541–2 football leagues 98
multiplication and division 37 height data activity 543 fountain borders 341
simplest form (lowest height of a triangle 352 heights and percentiles 541–2
hidden treasure investigation hidden treasure 278–9
terms) 35 house of cards 245
French mathematicians 249 278–9 ICT activities 99
frequency density 528 highest common factor (HCF) 7 metal trays 386
frequency tables 505, 510 Hindu mathematicians 3 mystic rose 96–8
histograms 525–6 numbered balls 436
estimating the mean 507 painted cube 471
grouped 512–13 with varying class intervals plane trails 277–8
functions 239 527–8 primes and squares 98
composite 243 probability drop 498
exponential 201 house of cards investigation 245 reading ages 542–3
gradient function 222–3 Huygens, Christiaan 475 tennis balls 387
inverse of 241–2 hyperbolas 198, 199 tiled walls 342
quadratic 193–5 hypotenuse 394 towers of Hanoi 436–7
reciprocal 198 triangle count 471–3
recognising and sketching I irrational numbers 4–5, 8
isosceles triangles 283
206–14 ICT activities Italian mathematicians 439
angle properties of a circle
G 343 K
Buffon’s needle experiment
Gauss, Carl Friedrich 475 500 Khayyam, Omar 101
geometric sequences (exponential cone construction 388 kites 286
exponential equations 246
sequences) 160 height data 543 L
applications 161–2 inequalities 246, 279
gradient function triangle ratios 342–3 Laghada 3
determination from first trigonometric ratios 437 Laplace, Pierre-Simon 475
vectors 473 length of a line segment 268–9
principles 222–3
of a polynomial 223–4 image of an object 452
see also differentiation improper fractions (vulgar
gradients
of a curve 184–7, 201–2, fractions) 30, 31
indices (singular: index) 62
219–22
of a curve at a point 229–30 algebraic 122–5
distance–time graphs 174
of perpendicular lines 271–2
speed–time graphs 178
of a straight line 219, 252–5

546

linear equations 126–7 negative enlargements 463 shortest distance from a line Index
linear functions 199 negative indices 64, 123 to a point 423
net pay 73–4
graph sketching 207 nets 287 Persian mathematicians 101
linear inequalities 142–3 nth term of a sequence 154, 155 pi (π) 5
linear programming 150
line segments exponential sequences 160 area and circumference of a
with quadratic and cubic circle 356–7
length of 268–9
midpoint of 269 rules 156–9 pictograms 510
lines of best fit 520 numbered balls piece work 74
lines of symmetry 310 pie charts 513–15
local maxima and minima 235, investigation 436 planes of symmetry 310
number line, representation of plane trails investigation 277–8
236 points of inflexion 235
long division 34 inequalities 24–5 polygons 286–7, 325
long multiplication 33 numbers, types of 4–6
loss 75–6 numerator 30 irregular 329
lower bounds 17–18 sum of exterior angles 326
O sum of interior angles 325
calculating with 19 polynomials
lower quartile 534–5 obtuse-angled triangles 283 gradient function 223–4
lowest common multiple obtuse angles 282 see also quadratic expressions
opposite side, right-angled population density 57, 58
(LCM) 7 position vectors 445
lowest terms, fractions 35 triangles 394 positive correlation 519–21
ordering 24 powers 5–6, 10
M order of operations 26–7, 34 of two 163
order of rotational symmetry see also indices
magnitude of a vector 444 prime factors 6–7
maximum points 197 310 prime numbers 5
overtime 74 primes and squares
local 235, 236
mean 504–5 P investigation 98
principal 76–7
for grouped data 507 painted cube investigation 471 prisms 361
median 504–5 parabolas 193, 197, 199
volume and surface area
calculation from a cumulative see also quadratic functions 362–3
frequency diagram 532–3 parallel lines 282
probability 476–7
metal trays investigation 386 angles formed within 319 combined events 488–91
metric units 346–8 equations of 265 conditional 492–4
midpoint of a line segment 269 parallelograms 286 relative frequency 482–3
minimum points 197, 208 area 354 tree diagrams 488–9, 491,
Pascal, Blaise 249, 475 494
local 235, 236 percentage increases and using Venn diagrams 480, 493
mirror lines 452
mixed numbers 30, 31 decreases 46 probability drop investigation
mode 504–5 percentages 32, 44 498
modelling, stretching a spring
discounts 75 profit 75–6
246 fraction and decimal proper fractions 30
money proper subsets 87
equivalents 43 proportion
currency conversions 73 interest 76–7
earnings 73–4 one quantity as a percentage direct 50–1, 166–8
interest 76–80, 161–2 inverse 53–4, 168
profit and loss 75–6 of another 45 pyramids
multiples 7 profit and loss 76 surface area 376
multiplication of a quantity 44–5 volume 374
of fractions 37 reverse percentages 47–8 Pythagoras 281
long 33 percentiles 534 Pythagoras’ theorem 5, 401
mystic rose investigation 96–8 height charts 541–2 in three dimensional objects
perimeter 351
N perpendicular bisectors, equal 426–7, 430

natural numbers 4 chords 313
negative correlation 521 perpendicular lines 282

gradients 271–2

547

Index Q rotational symmetry 310–11 multiplying equations 132–3
rotations 455, 457 with one linear and one non-
quadratic equations rounding 13
completing the square 140 linear equation 141
graphical solution 195–6 upper and lower bounds 17–18 substitution method 131–2
graph of completed square Rudoxus 281 sine curve 408–9
form 197 sine (sin) ratio 397–8
solving by factorising 137–8 S sine rule 418
speed 56–7, 58, 173
quadratic expressions, scale drawings 289–90 from distance–time graphs
factorising 110–11 scale factors
174, 176, 184
quadratic formula 140 of enlargements 460, 461 speed–time graphs
quadratic functions 193–5, 199 of similar figures 293
scalene triangles 283 acceleration and deceleration
graph sketching 207–9 scatter diagrams 519–20 177–8
turning points 197 sectors 285
quadrilaterals 285–6 area 368 area under 180–1
angles in 323–4 segment of a circle 285 non-linear 185–6
quartiles 534–5 selling price 75 spheres
semicircle, angle in 327 surface area 373
R sequences 153 volume 371
arithmetic 153–5 spread of data 504–5
radius 285 combinations of 163 square numbers 5–6, 163
range of data 504–5, 534–5 exponential (geometric) square roots 9–10
ratio method, direct proportion irrational numbers 5, 8
160–2 squares (quadrilaterals) 285
50, 51 formulae for the nth term standard form (scientific
rational numbers 4, 8
ratios of 154 notation)
key sequences 163 negative indices and small
dividing a quantity in a given with quadratic and cubic
ratio 52 numbers 67–8
rules 156–9 positive indices and large
increase and decrease by a term-to-term rules 153
given ratio 55 set notation 480 numbers 65–6
sets 85 stationary points 234–6
reading ages investigation complement of a set 87–8 statistics, history of 503
542–3 empty set 86 stem-and-leaf diagrams
intersection of 88–9
real numbers 5 problems involving 92 511–12
rearranging formulae 105–6, subsets 86–7 back-to-back 512
union of 89 straight line
112–13 universal set 87–8 equation of 257–63
reciprocal functions 198, 199 Venn diagrams 88–9 gradient of 252–5
short division 33 straight-line graphs 266
graph sketching 211–13 side–angle–side (SAS) rule, stretching a spring 246
reciprocals 5, 37 subsets 86–7
rectangles 285 congruent triangles 304 substitution 105
side–side–side (SSS) rule, subtraction
perimeter and area 351 of fractions 36
recurring decimals 39–40 congruent triangles of vectors 442
303, 305 supplementary angles 333
conversion to fractions 40–1 significant figures 14 surface area
reflections 452–3 similar shapes 284, 287, 293 of a cone 382
reflective symmetry 310 area and volume 297 of a cuboid 359–60
reflex angles 282 simple interest 76–7 of a cylinder 359–60
relative frequency 482–3 simplest form of fractions 35 of a prism 362–3
reverse percentages 47–8 simplifying algebraic fractions of a pyramid 376
rhombuses 285 114–15, 118 of a sphere 373
right-angled triangles 283, 394 simultaneous equations surveys 517
elimination method 130–1 Swiss mathematicians 391
Pythagoras’ theorem 401 graphical solution 267–8 symmetry 310–11
see also trigonometry
right angle–hypotenuse–side

(RHS) rule, congruent
triangles 305
right angles 282
roots 5, 8, 9–11

548

T triangles 283 V Index
angles in 321
tally charts 510 area 352, 423 vector geometry 446
tangent curve 411 congruent 284, 303–5 vectors
tangent (tan) ratio 394–5 construction of 288–9
tangents, equation of 233 ICT activity 342–3 addition and
tangents to a circle 315 similar 284, 293 subtraction 442

angle between a tangent trigonometric equations 412 ICT activity 473
and a radius 328 trigonometry 394 magnitude of 444
multiplication by a scalar
tennis balls investigation 387 angle between a line and a
terms of a sequence 153 plane 429–31 442–3
term-to-term rules for position vectors 445
area of a triangle 423 translations 440–1
sequences 153, 155 cosine (cos) 399–400 Venn diagrams 88–9
Thales of Alexandria 281 cosine curve 410 probability calculations
tiled walls investigation 342 cosine rule 420
time 83, 173 ICT activity 437 480, 493
time and a half 74 sine (sin) 397–8 vertex of a triangle 352
towers of Hanoi investigation sine curve 408–9 vertically opposite
sine rule 418
436–7 tangent (tan) 394–5 angles 319
transformations tangent curve 411 Villani, Giovanni 503
in three dimensions 426–7 volume
combinations of 465–6 turning points 197, 208
enlargements 460–1, 463 two-way tables 488 of a cone 377–9
reflections 452–3 of a prism 362
rotations 455, 457 U of a pyramid 374
translations 440–1, 458 of a sphere 371
translations 440–1, 458 union of sets (∪) 89 volume factors 297
trapeziums 286 unitary method, direct vulgar fractions (improper
area 354, 355
travel graphs proportion 50, 51 fractions) 30, 31
distance–time graphs 174, units
X
176 converting from one unit to
non-linear 183–7 another 347–9 x-axis 250
speed–time graphs 177–8,
metric 346–8 Y
180–1 universal set 87–8
tree diagrams 488–9, 491, 494 upper bounds 17–18 y-axis 250
triangle count investigation
calculating with 19 Z
471–3 upper quartile 534–5
triangle numbers 163 zero index 63, 123

549

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