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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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Topic based approach brings mathematics alive

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

lines labelled a b c and d

Introduction by determining the exact distance from 6

a number of satellites in orbit above

When you use the satellite navigation 4

system in a car or search on your

phone for the nearest branch of a shop

your electronic devices are using GPS

What s the point

Coordinates allow you to specify the exact 0 a b c d

coordinates The GPS system uses

position of any point on the Earth using

coordinates which are expressed in terms

just a pair of numbers

of latitude and longitude and are fixed

Starter problem y

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Investigate where she could draw the third point 7

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Objectives 3

By the end of this chapter you will have learned how to 2

Identify and plot coordinates in all four quadrants 1

Construct and interpret line graphs in context 0 1 2 3 4 5 6 7 8 9 10

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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

their pictures and self portraits

160 Geometry Transformations and symmetry SEARCH 161

1113 1115 1127

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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

height is measured If the base is not obvious turn the page round

and look at the triangle from a different direction 9 6 cm

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

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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

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