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International Mathematics Inte Witra Publishing Group

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835618_4.2.indd 4 23/06/16 1:57 AM 6 The ages (in years) of the first 30 people visiting the Tuileries Gardens in Paris on a certain day are shown. 25 65 34 48 4 55 32 45 23 43 23 37 45 36 26 39 43 45 29 15 15 20 64 37 61 25 17 23 45 36 a Construct a grouped frequency table for this data. b Construct a cumulative frequency curve.



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S TAT IS T ICS AND PR OB ABILIT Y
Reflect and discuss 1
ATL Example 1 What does cumulative frequency represent
The lengths of 50 Eelgrass leaves were measured The results are given Why can t you find the exact value of the median from a grouped
below to the nearest tenth of a centimeter Tip frequency table
Measuring a leaf How would you estimate the range from a grouped frequency table
46 3 35 6 75 2 43 7 49 0 42 6 47 1 50 3 50 7 55 6 gives continuous Estimate the range of lengths of Eelgrass leaves
56 0 52 7 57 3 57 3 34 2 58 0 37 0 59 3 68 1 59 9 data this has to
53 6 34 0 74 4 70 9 44 2 41 3 64 9 39 3 44 2 68 3 be rounded to a
certain degree of The class width is the difference between the maximum and the
54 2 40 7 76 8 38 2 70 3 33 5 64 3 64 6 68 7 69 9
accuracy to be minimum possible values in a class interval The class 18 x 20
48 9 51 6 51 7 46 0 52 0 45 6 64 1 63 4 45 1 62 0 used in statistical has class width 2
analysis because 20 18 2
a Construct a grouped frequency table to represent the data
An estimate for the range from a grouped frequency table is upper bound
b Find the class interval that contains the median of highest class interval lower bound of lowest class interval
c Write down the modal class
a minimum 33 5 cm maximum 76 8 cm When the class widths are equal the modal class is the class containing
range 76 8 33 5 43 3 cm the most data It has the highest frequency
43 3 Choose a class width to give between 5 and For grouped data you can find the class interval that contains the median
4 33 so use a class width of 5 cm Grouped data has a
10 15 equal width classes In this case 10 Add a cumulative frequency column to the frequency table to find which modal class instead
class interval contains the n 2 1 th value of one mode value
Length x cm Freq Cumulative Write the class intervals using inequality notation
Practice 1
30 x 35 3 3
1 The table shows the heights of a group of students on their 14th birthdays
35 x 40 4 7
40 x 45 6 13 Height x cm Frequency
45 x 50 7 20 Add a Cumulative frequency column to help 1 20 x 1 30 4
50 x 55 8 28 find the class interval that contains the median 1 30 x 1 40 6
55 x 60 7 35 1 40 x 1 50 8
The cumulative frequency is the sum
60 x 65 6 41 of the frequencies for every class up to 1 50 x 1 60 6
65 x 70 4 45 and including the current one Here the
cumulative frequency is 3 4 6 13 1 60 x 1 70 6
70 x 75 3 48
a Write down the modal class
75 x 80 2 50 The 25th and 26th values are in this class b Find the class interval that contains the median
b n 50 so the median is the n 1 50 1
25 5th data value 2 Here are the times taken for some students to complete a 1500 m race
2 2 measured to the nearest whole second
The class interval that contains the median is 50 x 55
The modal class is the class interval 325 580 534 500 532 328 600 625 450 435
c The modal class is 50 x 55
with the highest frequency 450 340 357 370 401 456 388 626 532 399
a Construct a grouped frequency table for this data Use class widths of
equal size and first class interval 300 x 350
b Find the modal class
c Determine which class interval contains the median value
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S TAT IS T ICS AND PR OB ABILIT Y
Reflect and discuss 4
6 The ages in years of the first 30 people visiting the Tuileries Gardens in
ATL If a set of data is grouped in equal class width intervals
Paris on a certain day are shown
Does the choice of class interval affect the estimate of the mean
25 65 34 48 4 55
What about the median class and modal class Explain
32 45 23 43 23 37
45 36 26 39 43 45 Does the choice of class interval affect the cumulative frequency curve
29 15 15 20 64 37 and box and whisker diagram Explain
61 25 17 23 45 36 Discuss the advantages and disadvantages of having too few classes
Do likewise for too many classes
a Construct a grouped frequency table for this data
b Construct a cumulative frequency curve
c Use your cumulative frequency curve to estimate the percentage of
visitors who are 25 or under
D Misleading statistics
How can real data ever be misleading
d Estimate the percentage of visitors who are older than 40
How do individuals stand out in a crowd
7 The cumulative frequency curves
give information about the ages Pedestrians
of pedestrians and pedal cyclists 450 Exploration 3
killed in Great Britain in 2014 400
The table gives the ages of fathers at the birth of their first child in
Cumulative frequency
a Draw a box and whisker Switzerland in 2013
diagram for the pedestrians 300
Age Freq Age Freq Age Freq Age Freq Age Freq
b Draw a box and whisker
200 12 0 22 388 32 5522 42 2203 52 189
diagram for the pedal cyclists
150 Use the same scale 13 0 23 598 33 5766 43 1773 53 146
Pedal cyclists
c Compare the ages of the and draw one
pedestrians and pedal cyclists
box and whisker 14 0 24 958 34 5612 44 1487 54 128
diagram above the 15 0 25 1414 35 5516 45 1187 55 95
other 16 2 26 1815 36 5225 46 856 56 86
10 20 30 40 50 60 70 80 90
Age 17 13 27 2365 37 4703 47 717 57 61
18 26 28 3073 38 4126 48 579 58 45
Exploration 2 Use the random 19 69 29 3792 39 3705 49 442 59 40
number button on 20 132 30 4382 40 3210 50 365 60 174
1 In your class record 500 random numbers between 1 and 30 generated
your calculator a 21 251 31 5108 41 2703 51 290
in the following way
spreadsheet
Use technology to produce random 3 digit numbers and record the last formula or random 1 Draw a grouped frequency table to represent this data Use equal class
two digits of each ignoring any over 30 number tables width intervals 1 x 10
For example You could find 11 x 20 etc Assume that all the fathers were under 70 What will
Construct a grouped frequency table with carefully chosen intervals As 15 random the highest class interval be
a class try different sized class intervals Use technology to calculate the numbers each and 2 Calculate an estimate for the mean and the five point summary of the
measures of central tendency combine them into age of fathers from your table
2 Compare your results with other students who chose different class one class set
3 Peter uses the class intervals below so that the frequencies in all the
intervals What are the differences
classes are roughly equal
3 Discuss how choosing different class intervals affects the mean median
class and modal class 0 x 25 26 x 29 30 x 34 35 x 38 39 x 43 44 x 50
4 Why would it not be sensible to represent this data in a stem and leaf
diagram Construct a grouped frequency table using his class intervals
5 Represent the data in a cumulative frequency curve and box and Calculate an estimate for the mean and the five point summary from
whisker diagram this new table
6 Compare your results with others
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6 Graphs Check in
1 a Draw a number line from 10 to 10
b Use arrows to mark these points
i 2 ii 7 iii 4 iv 9 v 0
2 Write down the heights of the
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Introduction by determining the exact distance from 6
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When you use the satellite navigation 4
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Coordinates allow you to specify the exact 0 a b c d
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just a pair of numbers
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Starter problem y
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Objectives 3
By the end of this chapter you will have learned how to 2
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9a Transformations 5
Exercise 9a
1 Copy each diagram on square grid paper 3 On square grid paper draw the x axis
Transformation Reflect the shape in the mirror line Give from 0 to 15 and the y axis from 0 to 10
A transformation moves
the name of the completed shape and a Plot each shape on the same grid
a shape to a new position
state if it is regular equal sides and Shape A 3 7 4 8 2 10 1 9
Here are some different
equal angles and 3 7
transformations
a a b b c c d d Shape B 0 5 1 4 3 6 2 7
Object Image
Shape C 5 6 4 7 3 6 and 5 6
Shape D 5 1 4 3 2 3 1 1
A reflection flips an object over a mirror line a a b b c c d d Shape E 4 3 6 5 5 6 3 4
You describe a reflection by giving the mirror line Shape F 6 7 7 8 5 10 4 9
Object Image
A rotation turns an object about a point called the Mirror line b Give the mathematical name of each
centre of rotation 2 Copy this diagram shape
a Rotate the isosceles right c Translate each shape on the same grid
You describe a rotation by giving Image angled triangle through Shape A 6 units to the right and
the centre of rotation 180 about the midpoint of 0 units up
the angle of rotation the longest side Shape B 10 units to the right and
the direction of turn clockwise or anticlockwise Object b Mark the equal angles and the equal 3 units up
sides on the completed quadrilateral Shape C 6 units to the right and
A translation slides an object c State the mathematical name of the 3 units up
quadrilateral Shape D 7 units to the right and
Image 2 units up
You describe a translation by giving the distance
Shape E 4 units to the right and
moved left or right then the distance moved up or 1
down Object 3 Shape F 6 units to the right and
3 units down
Draw the hexagon after a
y y Did you know
clockwise rotation of 90 Problem solving
about 0 1 4 Draw a rhombus like this on isometric
y paper Use a mirror to form these
shapes Draw diagrams to illustrate the
x x position of the mirror in each case
3 2 1 0 1 2 3 2 1 0 1 2
a b c d e f g
Use tracing paper to
x It is thought that many of the Old
3 2 1 0 1 2 rotate the shape
Master painters used mirrors to paint
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160 Geometry Transformations and symmetry SEARCH 161
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5 Sets Write each of the following sets as a list of members
1 the days of the week
2 the even numbers between 1 and 11
3 the prime numbers between 10 and 20
Thirty students wrote down what kind of transport they used
4 the multiples of 3 between 1 and 20
one day to get to school
5 the last four letters of the Roman alphabet
15 used a bus 18 used a car
6 used both a bus and a car 6 multiples of 5 between 20 and 50 inclusive
How can you nd the number that used neither a bus nor a car 7 the prime factors of 70
You should be able to solve this problem after you have worked 8 the square numbers between 1 and 100 inclusive
through this chapter 9 the triangular numbers between 1 and 50 inclusive
10 the different factors of 12
Class discussion
Thirty students were asked if they had a pencil with them They were then
Finite and infinite sets
asked if they had a pen with them When we need to refer to a set several times we can use a capital letter to label that set
For example
25 students said they had a pencil
21 students said they had a pen A months of the year beginning with J
or A January June July
These two numbers add up to more than the number of students in the class
In many cases it is not possible to list all the members of a set For example if
Why do you think this is the case
N positive whole numbers
Set notation we can write
A set is a clearly de ned collection of things that have something in common for example N 1 2 3 4 5
a set of drawing instruments a set of books where the dots show that the list continues inde nitely
Things that belong to a set are called members or elements of a set These members are N is an example of an in nite set whereas A is an example of a nite set
usually written down separated by commas and enclosed in curly brackets
For example the set of prime numbers between 0 and 10 can be written as 2 3 5 7
We do not have to list all the members of a set when we can use words to describe them Exercise 5b
For example instead of 1 2 3 4 5 19 20 we can write whole numbers from 1 to 20
inclusive State whether the following sets are nite or in nite
1 A vowels of the Roman alphabet
Exercise 5a 2 B the factors of 30
3 C even numbers
4 D even numbers between 3 and 9
Worked example 5 E students taking violin lessons
Write the prime factors of 30 as a list of members 6 F odd numbers
30 2 15 7 G grains of sand on the Earth
2 3 5 8 H multiples of 3
Therefore the prime factors of 30 2 3 5 9 I different shapes of triangle
10 J different languages spoken in the world
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STP Maths 8 7 Areas of triangles and parallelograms
The base of the triangle is the side from which the perpendicular 19 22 20 cm
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4 cm 16 cm 12 cm
Worked example
Find the area of the triangle
8 cm 14 cm
Look at this diagram from 15 cm
the direction of the arrow
Area 12 base height
10 cm 1 10 8 cm2
40 cm2 21 24 4 2 cm
12 cm 12 cm
Find the areas of the triangles
13 16 12 cm
11 cm 5 12 cm
16 cm For questions 25 to 30 use squared paper to draw axes for x and y from 0 to 6
using 1 square to 1 unit Find the area of each triangle
25 ABC with A 1 0 B 6 0 and C 4 4
14 17 15 cm 26 PQR with P 0 2 Q 6 0 and R 6 4
27 DEF with D 1 1 E 1 5 and F 6 0
10 cm 28 LMN with L 5 0 M 0 6 and N 5 6
8 cm 29 ABC with A 0 5 B 5 5 and C 4 1
30 PQR with P 2 1 Q 2 6 and R 5 3
15 30 cm 18
31 Find the area of each triangle in this diagram The squares represent
square centimetres Explain your answers
20 cm a b c d e f
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Exercise 5
1 Find the equations of the lines A and B 2 Find the equations of the lines C and D Substitute for y and x 2 1 2 c
The equation of the line is y 1 x 8
2 Exercise 6
2 Find the equation of the line which
1 passes through 3 4 and is perpendicular to y x 2
3 Look at the graph 1
2 passes through 2 1 and is perpendicular to y 2 x 1
a Find the equation of the line which is parallel to line A and
A 3 passes through 1 6 and is perpendicular to y 4x 3
which passes through the point 0 5 6 1
b Find the equation of the line which is parallel to line B and 4 passes through 5 0 and is perpendicular to y 6 x 4
which passes through the point 0 3 4
5 passes through 7 2 and is perpendicular to y 5x 2
4 Look at the graphs in Question 1
7 4 Plotting curves
a Find the equation of the line which is parallel to line A and 2
which passes through the point 0 1
b Find the equation of the line which is parallel to line B and Example y
which passes through the point 0 2 Draw the graph of the function y 2x2 x 6 for 3 x 3
a x 3 2 1 0 1 2 3 y 2x2 x 6
Example 12
Find the gradient of lines A and B y 2x2 18 8 2 0 2 8 18 10
A x 3 2 1 0 1 2 3
Gradient of line A m1 2 8
1 6 6 6 6 6 6 6 6
Gradient of line B m2 2 4 6
y 9 0 5 6 3 4 15
Lines A and B are perpendicular to each other Notice that the 4
product of their gradients m1 m2 1 2 b Draw and label axes using suitable scales 2
This is true for all pairs of perpendicular lines and hence we can c Plot the points and draw a smooth curve through them with
3 2 1 0 1 2 3 x
find the gradient of one when we know the other 2 0 2 4 x a pencil 2
Find the equation of the line perpendicular to y 3x 5 which d Check any points which interrupt the smoothness of the curve 4
passes through 2 2 e Label the curve with its equation 6
Gradient of perpendicular line m2 1 3 3
so we may write the equation as y 3x c
242 Graphs Plotting curves 243
Complete Mathematics
837835 Ch07 indd 242 243 0580 Extended for Cambridge IGCSE Student Book Fourth edition 16 06 16 2 04 PM
Achieve top marks with exam practice
13 3 1 The maximum 13 3 4 Identification techniques
0 At the maximum the gradient is zero That s how we find them Technique 1 We could find the value of the gradient to the left and
To the left of the maximum the gradient is positive right of the stationary point
To the right of the maximum the gradient is negative
Example 13 2
If we draw the graph of the gradient of the curve y f x
it must cross the x axis at the stationary value being positive Find the nature of the stationary points of the following graph
to the left of that value and negative to the right y x3 3x2 9x 4
The gradient of the gradient graph f x f x must be negative
Step 1 Find the stationary points
These are 1 9 and 3 23 From Example 13 1
The gradient function is
3x 2 6 x 9
13 3 2 The minimum dx
At the minimum the gradient is zero but the gradients Step 2 We test the gradient for values of x close to the stationary points
are opposite that when the stationary point is a maximum
To the left of the minimum the gradient is negative dy
To the right of the minimum the gradient is positive dx
0 If we draw the graph of the gradient of the curve y f x 1 1 1 23 Actual gradient values do not matter
it must cross the x axis at the stationary value being negative 0 9 1 17 Maximum We just want to know whether they
are positive or negative
to the left of that value and positive to the right
The gradient of the gradient graph f x f x must be positive Minimum
The point 1 9 is a maximum and the point 3 23 is a minimum
Technique 2 We find the value of the second derivative at the stationary point
13 3 3 A stationary point of inflexion
At a stationary point of inflexion the gradient is Example 13 3
zero but the gradient is either positive on both Find the nature of the stationary points of the following graph
0 0 sides or negative on both sides of the stationary y x3 3x2 9x 4
If we draw the graph of the gradient of the curve Solution
y f x it must touch the x axis at the Step 1 Find the stationary points
stationary value These are 1 9 and 3 23
x From Example 13 1
The graph will have either a maximum or a The gradient function is
minimum at this point dy
3x 2 6 x 9
y f x y f x The gradient of the gradient graph dx
x f x f x must be zero
192 Applications of the derivative Identifying the nature of stationary points 193
Additional Mathematics for Cambridge IGCSE O Level Student Book
www oxfordsecondary com igcsemaths 13


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