numbers and space

Transcription

numbers and space

numbers and space
first print
first edition, 2003
L.A. Reichard
S. Rozemond
J.H. Dijkhuis
C.J. Admiraal
G.J. te Vaarwerk
J.A. Verbeek
G. de Jong
N.J.J.M. Brokamp
H.J. Houwing
R. de Vroome
J.D. Kuis
F. ten Klooster
F. van Leeuwen
S.K.A. de Waal
J. van Braak
This book can lead to flashes of insight.
Cover design: In ontwerp, Assen, Netherlands, in collaboration with GREET, Amsterdam, Netherlands
Basic book design: Gerard Salomons BNO, Groningen, Netherlands
Layout: Grafikon, Bruges, Belgium
Translation: Overtaal BV, Utrecht, Netherlands
© 2002 EPN, Houten, Netherlands
ISBN 90 11 07708 3
All rights reserved. None of the contents of this publication may be copied, stored as a digital file, or published in any
way, shape or form, whether electronically, mechanically, by xerography, photography, or by any other means,
without the prior written consent of the publisher.
As far as copies of this publication are allowed to be made according to Article 16B of the 1912 Copyright Act in
conjunction with the Decree of 20 June 1974, Official Gazette 351, as amended by the Decree of 23 August 1985,
Official Gazette 471, and Article 17 of the 1912 Copyright Act, the legally stipulated reimbursements are to be paid to
the Stichting Reprorecht (P.O. Box 3060, 2130 KB Hoofddorp, the Netherlands). The publishers must be contacted
for permission to publish part (or parts) of this publication in the form of anthologies, readers or other compilations
(Article 16, 1912 Copyright Act).
Preface
Dear Teacher,
Parts 1 havo/vwo part 1 and 1 havo/vwo part 2
This is the English language version of getal en ruimte 1 havo/vwo deel 1.
Parts 1 havo/vwo part 1 and 1 havo/vwo part 2 are intended for first-year havo/vwo pupils.
We have conceived these completely revised first-year volumes in recognition of the demand for
customised books. For this reason, there are four parallel versions of the 1 havo/vwo parts, namely
1 vmbo-T/havo parts, 1 vwo parts and two language versions of the 1 havo/vwo parts 共English and
Dutch兲. There is a great deal of similarity between the 1 vwo- and the 1 havo/vwo parts; however, there
is a difference in complexity. Simpler exercises have been excluded from the 1 vwo parts and replaced
by more difficult ones. Here and there, extra paragraphs have been added to the 1 vwo part.
The material is more compact compared to the previous first-year parts, resulting in a reduction in the
number of chapters from twelve to ten. Due to the more compact structure, pupils will need to spend
less time on the simpler subjects.
The amount of material is based on four lessons per week. Each of the first-year parts contains five
chapters. Vwo pupils complete the basic education programme within two years.
ICT
From the outset, an important criterion for the writing of these two parts was to further integrate ICT.
The CD-ROM included with the workbooks therefore plays an important role. However, a definite
choice was made in favour of a two-pronged approach: working only from the books is not a problem,
the CD-ROMs merely provide a different route.
The integration of ICT focuses on the following criteria:
• Pupils must be able to work independently with ICT. The CD-ROMs provide pupils with the
software needed for working at home.
• Teachers can use the books in their own way, with or without the CD-ROMs.
The BRE model
The 1 havo/vwo parts have been structured according to the Basic-Revision-Extra material model,
where pupils take a diagnostic test after completing the basic material. What is new is that the
diagnostic test appears immediately after the summary, thereby accentuating the importance of
diagnostic testing. The last chapter contains a selection of combined exercises in chapter order. The
basic material is differentiated, recognisable by the ‘open’ exercise icon. This material is primarily
destined for future vwo pupils.
Independent study
In principle, pupils can work through the material independently. Due to the new design, pupils will
quickly recognise what they need to concentrate on. Examples, for instance, are shown on a shaded
background, and exercises are marked according to type, such as orientation or final. These elements
carefully prepare pupils for independent study in the second phase. For more details concerning the
working and teaching methods, as well as the choice of subject matter, we refer to the manual
accompanying the 1 havo/vwo parts.
We would very much appreciate users’ comments.
Spring 2002
How the book is structured
This section devotes specific attention to
skills.
Index
Combined exercises
IMA
Chapter 5
Chapter 4
Combined exercises per chapter and level,
wich are an excellent examination preparation.
This lists the pages on wich a term first
appears.
}
Skills
Chapter 3
Chapter 1
Chapter 2
Preface
Contents
Integrated Mathematical Activities.
Chapter intro
An attractive introduction to the subject
matter, together with a list of learning
objectives, materials, and a possible ICT
route.
Paragraph
See facing page.
Extra material
Summary
Revision
Diagnostic test
Computer paragraph
The last paragraph of some chaptes is to
be carried out on a computer.
Summary
Paragraphs
Computer paragraph
An overview of the chapter’s theory elements.
Diagnostic test, Revision, Extra
material
The diagnostic test is used to check
whether the basic material has been
understood. Each test question refers to
one or more exercises in the Revision section. Once the basic material has been
mastered, the Extra material section provides further in-depth training.
The diagnostic test can also be carried out
on a computer, using the CD-ROM
attached to the workbook.
Structural elements of a paragraph
Each paragraph features a careful escalation of orientation, theory and assimilation.
O 47
Orientation exercises
•
Theory sections are preceded by orientation exercises.
48
Standard exercises
•
49
Differentiation exercises
•
A 50
Final exercises
•
Theory
•
Digital learning line
•
Theory is followed by standard exercises to ensure
proper assimilation.
Every so often, there are exercises which require a
larger mental step. This is to allow differentiation
according to level.
The final exercises contain the essence of a paragraph
and indicate the desired level of competence.
The theory elements consist of the most important
information in the form of definitions, clear examples
and application methods.
Certain theory elements and the accompanying
exercises can be replaced by ICT. The software required
can be found on the CD-ROM in the workbook.
Workbook references
•
Theory A
C
51
•
䉴workbook
Defenitions
These contain the most important theory
aspects.
Some exercises refer to drawings in the workbook.
Occupation block
T H E O C C U PAT I O N
Example
Orientation for study and occupation.
Where would you come across this
aspect of mathematics in practice?
Here the theory is explained by way of a
sample exercise, often accompanied by
notes.
History
Working method
The standard approch or sequence of
steps applicable to certain exercises or
problems.
Mathematics was discovered by people,
often involving interesting historical
aspects.
Rule
This is a rule applied in mathematics.
This is an important sentence.
To the pupil,
Contents
1
2
3
1
Space shapes
1.1
1.2
1.3
1.4
1.5
1.6
Lines of vision
Shapes
Cubes and squares
Cuboids and rectangles
Cylinders and circles
Summary
Diagnostic test
Revision
Extra material
4
7
10
17
21
24
26
28
30
2
Numbers
32
2.1
2.2
2.3
2.4
2.5
2.6
2.7
Arithmetic
Decimal numbers
Fractions
The calculator
Arithmetic in everyday situations
Ratios
Summary
Diagnostic test
Revision
Extra material
34
38
42
48
52
56
60
62
64
68
3
Locating points
3.1
3.2
3.3
3.4
3.5
3.6
Where are you
Positive and negative numbers
Axes
Adding and subtracting positive numbers
Adding and subtracting negative numbers
Summary
Diagnostic test
Revision
Extra material
Skills
2
70
72
75
78
86
91
94
96
98
101
104
4
Diagrams
108
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Pictograms and bar charts
Rises and falls
Drawing graphs
Graphs and relationships
Periodic graphs
Diagrams
Diagrams using the Dutch Railways 共NS兲 route planner
Summary
Diagnostic test
Revision
Extra material
110
113
117
122
124
127
135
136
138
142
146
5
Lines and angles
150
5.1
5.2
5.3
5.4
5.5
5.6
Lines
Angles
Measuring angles
How to draw angles
Wire models and intersecting lines
Summary
Diagnostic test
Revision
Extra material
152
158
162
169
172
178
180
182
186
IMA Mathematics and art
190
Combined exercises
198
Glossary
214
Index
218
Illustrations
221
4
5
chapter
01
Space
The watermelons shown in the photo come from Japan. These
sorts of melons are grown in glass pots. They are intended for
export. At the auction melons like these are sold for around
€ 100.
• What shape do you think the glass pot is in which they are
grown?
• State one advantage of the melons having this shape.
• Are there any disadvantages?
• What shape is a normal watermelon?
2
Chapter 1
what you can learn
ICT
–
–
–
–
– Exercises: introduction to the subject of cd-rom.
Exercises on page 6.
– Learning: digital learning line on pages 11, 12, 13
and 14 with the applets Results and Draughtboard.
– Exercises: exercises accompanying each
paragraph.
– Testing: diagnostic test.
What are space figures?
What do space figures consist of?
How do you make a cut-out of a space figure?
How do you use your pair of compasses?
what you need
– workbook and cd-rom
– protractor triangle, coloured pencils, pair of
compasses, scissors, sticking tape and glue
– a roll of toilet paper, magazines or folders.
1.1 Lines of vision
O
1
Frits is in a hot air balloon.
He sees a farmer, a sheepfold and 15 sheep.
Refer to figure 1.1.
How many sheep can the farmer see?
Theory A
In figure 1.2 you can see a barn in a meadow, with
the view from above. 18 sheep are grazing nearby.
This view is known as a top view.
figure 1.1 What Frits can see from the hot air balloon.
figure 1.2 You can use lines of vision to determine what you can and cannot see.
The farmer is standing at point B. He is unable to see the sheep
behind the barn. To find out how many sheep he can see you have to
draw two lines. These lines are known as lines of vision. The farmer
cannot see the sheep between the red lines of vision, but he can see
those outside the lines. He can therefore see 12 sheep.
Lines of vision
A line of vision extends from your eye to the object you can
see. You can use lines of vision to indicate the limits of the
area you can see.
4
Chapter 1
2
Chickens are scratching around in the farmyard.
Mirjam is standing in front of a wall over which
she is unable to see.
Figure 1.3 shows you the top view. This figure is
.
featured twice in your
Use the first figure for questions a and b and the
second figure for questions c and d.
a Mirjam is standing at point A. In order to
find out how many chickens she can see you
have to draw two lines of vision in your
figure 1.3 The two walls mean that you are unable to see
workbook. Do that now.
all the chickens, no matter where you are on the path.
How many chickens can she see?
b Paul is standing at point B.
How many chickens can he see?
c Jeroen is walking along the path. In the second
figure draw point C to show exactly where
he must stand in order to see two chickens.
d Colour the section of the path red from which
you would be unable to see any chickens at all.
3
Below is a photo of a stately home.
In the garden there is a sundial on a pillar.
D
1
house
C
sundial
A
B
2
figure 1.4 A photo of a stately home and a top view.
a In the photo, the sundial appears to be bigger than the stately
home. How did the photographer manage to create this
impression?
b Next to the photo you will see a top view of the garden and
the house. Where was the photographer standing when he took
the photo? Choose A, B, C or D.
c From which point would the photographer be unable to get
both the sundial and the building in one and the same photo?
Choose A, B, C or D.
Space
5
4
Captain Mossel’s fishing boat is sailing around
the three islands Alba, Basta and Calei.
a Which island does captain Mossel see in the
middle at 14.00 hrs?
b At which two moments does captain Mossel
see the island of Calei in the middle?
c At a certain point in time, captain Mossel can
see the islands as shown in the smaller picture
at the bottom. At about what time is that?
figure 1.5
A
5
Seven children are playing hide and seek.
Six have hidden behind walls. Bart has to try and find them.
a How many children can Bart see if he stands next to the oak tree?
figure 1.6 Bart stands next to the oak tree and looks whether he can spot any of the children.
b Bart walks to the beech tree to see if anyone has hidden there.
On his way, he looks around in all directions. Which of the
children can he then see?
c Bart returns first to the oak tree and then goes to the chestnut
tree. On his way, he has a good look round again.
Who can he see now?
d Who has found the best hiding place?
6
6
There is a CD-ROM which goes with the
.
You can now do the exercises in paragraph 1 on the CD-ROM.
Chapter 1
1.2 Shapes
O
7
The illustration alongside is of Nol’s desk.
On the desk are all kinds of objects.
a Together with your other classmates, collect
objects which have the same form as the items
shown in the picture.
b Divide the objects into groups. Pay attention
to the material they are made from. What are
the groups you have made?
c When dividing the objects into groups you
can also take other characteristics into
account. Divide the objects into groups in two
other ways. Which characteristics did you
take into account?
figure 1.7
Theory A
Nol’s things all take up space.
We refer to them as space figures. You can see space figures all
around you. Some space figures have special names. The following are
examples of the most common space figures.
figure 1.8 The most common mathematical space figures.
Space
7
8
Write down the mathematical names of the objects
on Nol’s desk.
9
a Which of the space figures in figure 1.8 only
have flat surfaces?
b And which have only rounded surfaces?
c And which ones can be rolled?
Mathematical figures
The pointed roof of the
10 Jeroen has left imprints of space figures in the
sand. The imprints are shown in the illustration
below. Write down the name of the space figure共s兲
that corresponds to each imprint.
merry-go-round.
The ice-cream and its cone.
The petticoat that Annabel found,
And the cylinder of a bone.
The tip of a multicoloured pen.
The headlight on your bike.
The pointing beak of a little wren
The road along a dike.
The list completely meets the rule.
But you know there’s nothing like
Geometry to make you look a fool.
Marjolein Kool
figure 1.9 Four imprints of space figures in the sand.
11 A lot of objects have been made from two or
more space figures.
Look at the photo of the Atomium, for example.
It was constructed for the World Exhibition
EXPO ’58.
a What space figures link the spheres?
b Patrick is looking at the Atomium and can
only see seven spheres.
Explain where he might be standing.
12 a From which two space figures is a pencil
made?
b State a couple of other objects that consist
of two or more space figures.
Explain which space figures these are.
figure 1.10 The Atomium is located in Brussels.
8
Chapter 1
13 The largest spherical building in the world is in Stockholm.
The Globe Arena is a cultural centre which is used for concerts,
circus shows, exhibitions and sports competitions.
figure 1.11 The Globe Arena at the Soderstadion in Stockholm. The Eurovision Song
Contest was held in this stadium in May 2000.
a You should be able to see a number of lines
on the sphere. Are these straight lines?
Explain your answer.
b It looks as if the lines on the sphere form
squares. But are they really squares?
A 14
a Marit has drawn straight lines on the curved
surface of a cylinder.
How has she done this?
b Are there any other space figures on which
you can draw straight lines on a curved
surface?
If so, which ones?
Write down how you would draw the line.
figure 1.12 This work of art is located in Cape Town.
Which space figures can you see?
Space
9
1.3 Cubes and squares
O 15
a How many photos can fit into the photo cube shown?
b What shape are the photos?
c Is it possible to hold the cube in such a way that you can see
two photos? If so, how should you hold it?
d Is it possible to hold the cube in such a way that you can see
four photos?
e How should you hold the cube so that you can see just one
photo?
figure 1.13 You cannot see all the
photos in a photo cube in one go.
O 16
Figure 1.14 shows a cut-out of a cube. A cut-out like this without
glue edges is called a net. The net can also be found in your
. The squares shown in the workbook are 4 by
4 cm.
a Cut out the net and fold it to make a cube.
b Paste glue on one of the squares and stick the net into your
exercise book.
figure 1.14 A net of a cube.
Theory A
You can also use other space figures to make a net. Below you can see
a pyramid and its net.
figure 1.15 A pyramid and its net.
Net
The net of a space figure is like a cut-out without adhesive
edges.
10
Chapter 1
17 Is figure 1.16 the net of a cube?
Try to work the answer out in your head first.
If you cannot work it out, re-draw the figure and cut it out.
figure 1.16
Everything next to the blue line can be replaced by ICT.
Use the
in the workbook.
18 Not all the following figures are nets of a cube.
From which figures can you, in fact, make a cube?
a
b
c
d
f
e
g
h
figure 1.17
Space
11
19 If you fold up the net in figure 1.18 you can make a cube.
Which two cubes shown in figure 1.19 can you make?
.
If necessary, use the net in the
figure 1.18
figure 1.19
Theory B
The net of a cube consists of six squares.
A square is a flat shape.
D
side CD
C
vertex C
Figure 1.20 shows a square marked by ABCD.
The corners or vertices are A, B, C and D.
Vertices are always given a capital letter.
All squares have four sides.
The sides of the square ABCD are side AB,
side BC, side CD and side AD.
The four sides of the square are equally long.
A
figure 1.20 Square ABCD has four equal sides.
20 a Draw a square with sides measuring 6 cm.
b Mark the vertices with the letters K, L, M and N. Mark
the bottom left-hand corner with a K, the bottom right-hand
corner with an L, the top right-hand corner with an M and
lastly the top left-hand corner with an N. You will now have
drawn the square marked by the letters KLMN.
c Write down the letters for each of the four sides of the square.
12
Chapter 1
B
21 a Figure 1.21a shows five squares.
Explain where the five squares are.
b How many squares can you see in figure 1.21b?
c And how many in figure 1.21c?
a
b
c
figure 1.21 Some squares overlap.
Theory C
The illustration below shows the cube ABCD EFGH
drawn on graph paper. The cube has six flat surfaces.
Those flat surfaces are called the faces of the cube.
Face EFGH is coloured blue.
The borders of the cube are called edges.
Graph paper is paper
covered in small
squares.
face EFGH
G
H
E
F
C
D
A
edge AB
B
vertex B
figure 1.22 Cube ABCD EFGH with vertices, faces and edges.
The edges are 4 cm long.
The vertex of the cube is where three edges meet.
In vertex A these are edge AB, edge AD and edge AE.
Space
13
22 a Which face of the cube in figure 1.22 is yellow?
b In the face ADHE you can see four edges.
Which edges are these?
c Which edges meet in vertex C ?
23 a Edge CG belongs to two faces.
Write down which two faces these are.
b H is a vertex of three faces.
Write down which three faces these are.
24 a Is face DCGH a square?
b Is face BCGF really a square?
c In the drawing of the cube, edge BC is shorter than edge
AB. In reality they are equally long.
Which other edges have been drawn shorter than they really
are?
25 a Copy the cube shown in figure 1.22 into your exercise book
exactly as shown. Mark the vertices with capital letters.
b How many edges are shown by dotted lines? Why do you
think this is?
c How many edges does a cube have?
d How many faces does a cube have?
e How many vertices does a cube have?
26 Study the comic strip below. Finish Els’s last sentence.
Listen Els!
What have you found
out this time, Bas?
A cube has 8 vertices and 3
edges meet in each vertex.
figure 1.23 Is Bas’s argument right?
14
Chapter 1
So what?
A handy way of counting
the edges of a cube.
So … 8 times 3
is 24 edges!
Oh yeah?
How then?
You can't fool me.
You have … all …
27 a Write down how many blocks are needed for each structure shown below.
a
b
c
d
figure 1.24 The blocks have edges measuring 1 cm.
b The blocks are sold in boxes of 64. A box costs € 4.95.
Jef wants to make all four of the structures shown in figure 1.24 at the same time.
How many boxes does Jef need? How much is that going to cost?
28 a The structure shown in figure 1.24c has to become a cube measuring 4 by 4 by 4 cm.
How many blocks do you need to expand figure 1.24c to make that cube?
b Jef is going to use the individual blocks shown in figure 1.24d to make as large a cube
as possible. How many blocks will he then have left over?
29 a Write down how many blocks you need for each structure shown in figure 1.25.
b The structure shown can be made using five layers.
How many blocks do you then need?
And how many do you need for seven layers?
a
b
c
d
figure 1.25 The first four structures of the series.
Space
15
A 30
Figure 1.26 shows a structure of blocks which are all glued
together. Piet is painting the structure.
He only paints the outside of the structure and not the underside.
a How many squares has Piet already painted red?
b How many squares does Piet still have to paint?
figure 1.26
A 31
Leanne wants to be an architect when she grows up. She already
likes using her box of blocks to design attractive models of flats
and office buildings.
Each cube shown in the design in figure 1.27 would, in reality,
have edges measuring 15 metres. The construction costs per
cube are € 20,000.
The land price is € 450 per m 2.
Calculate the total costs of the building.
figure 1.27 Design for the ‘Groenesteyn’ residential oasis.
16
Chapter 1
1.4 Cuboids and rectangles
O 32
Which of the objects shown below are in the shape of a cuboid?
figure 1.28
O 33
a In your
you will find a page showing the net of
a cuboid which you can cut out. Cut out the net and fold it to
make a cuboid.
b How many vertices does a cuboid have? And how many edges?
And how many faces?
c Write letters on the cuboid to mark the vertices. Write down each
letter three times as has been done in the photo with the letter F.
H
E
G
F
1,5 cm
D
C
2,5 cm
A
3 cm
B
figure 1.29
d Paste the net into your exercise book so that the letters remain
visible.
e Which face is the same size as the face ABCD?
f Which edges are just as long as edge EH ?
Space
17
34 a Copy the cuboid shown in figure 1.30 into
your exercise book.
b Colour the edges that meet at vertex E
green.
c Colour the face BCGF red. Which face
is the same size as the red face?
f Which edges are just as long as edge AE ?
e Which edges have been drawn shorter than
they really are?
H
G
E
F
5
C
D
35 a Tessa says, ‘I can make a cuboid with eight
edges of 5 cm and four edges of 7 cm.’
Is what Tessa says correct?
b Sandra says, ‘I can make a cuboid with six
edges of 5 cm and six edges of 7 cm.’
Is what Sandra says correct?
4
A
B
3
figure 1.30 The dimensions are in cm.
36 A cuboid has edges measuring 6 cm, 4.5 cm
and 3 cm.
a Draw the net of this cuboid on a separate
piece of graph paper. Start with the red
face.
b Cut out the net and make a cuboid.
3 cm
4,5 cm
6 cm
figure 1.31
37 The figures below are not all nets of a cuboid.
Have a look to see which figures you can use to make cuboids.
a
figure 1.32
18
Chapter 1
b
c
d
Theory A
6
7
A rectangle is a quadrilateral with four right angles.
160
20
40
140
50
130
8
10 0
0
90
17
10 0
3
5
2
4
Rectangle
ZO
ET
1
ER
M
3
EE
R
70
110
2
60
120
1
0
3
1
2
2
1
30
150
3
20
160
4
5
10 0
17
6
7
The net of a cuboid consists of six rectangles. The drawing alongside
is that of a rectangle.
One of the angles of your protractor triangle fits exactly into one of
angles of the rectangle. An angle like this is called a right angle.
150
30
140
40
130
50
120
60
110
70
0
10 0
8
figure 1.33 A rectangle has four right
angles.
The above-mentioned definition is the mathematical definition of a
rectangle. As you can see, it does not say anything about the length of
the sides. The four sides may therefore be of equal length.
In rectangle PQRS the diagonal PR has been drawn.
S
R
The word diagonal comes from Greek and
means: drawn obliquely.
al
on
g
dia
P
Q
figure 1.34 One of the two diagonals
is shown in this rectangle.
38 a Draw a rectangle ABCD whereby
AB ⫽ 4 cm and AD ⫽ 3 cm.
b Which side is just as long as side AB?
c Draw the diagonals AC and BD.
Are they equally long?
D
39 Figure 1.35 includes a drawing of part of the
rectangle ABCD.
a Copy the figure into your exercise book.
b Check using your protractor triangle that you
have drawn a right angle at vertex A.
c How many mm long is side AB?
And side AD?
d Complete the rectangle.
Add the missing letter.
A
B
figure 1.35
Space
19
40 In figure 1.36 KL is a side of the square
KLMN. Copy this figure and complete the
square.
A 41
a Jorien says: ‘All squares are rectangles.’
Do you agree with Jorien?
b Femke says: ‘All rectangles are squares.’
Do you agree with Femke?
c Is every cuboid a cube or is every cube a
cuboid? Explain your answer.
L
K
figure 1.36
These are all rectangles.
A 42
20
a There are a number of ways of making a rectangle from 12
whole grid squares.
Draw all the possibilities.
b How many different rectangles can you make from 24 grid
squares?
And how many can you make from 48?
And how many can you make from 96?
c State how may rectangles you can make from the following
numbers of grid squares.
7
15
32
49
100
101
Chapter 1
1.5 Cylinders and
circles
O 43
a How would you describe the shape of a tin?
b How many flat faces are there?
What shape are those faces?
c How many curved faces are there?
figure 1.37
O 44
An empty roll of toilet paper has the shape of a
cylinder. There are no flat faces.
a Cut a roll open lengthways and lay it down
flat.
b What is the flat shape that you can now see?
O 45
a On a separate sheet of paper, draw two
rectangles measuring 8 by 15 cm and cut them
out.
b Make a cylinder from each rectangle. Make
sure you end up with two different cylinders.
O 46
The net of a cylinder consists of two circles and
a rectangle.
and
a Cut out the net in your
make a cylinder.
b Paste glue on one of the circles and stick the
net into your exercise book.
figure 1.38
Theory A
A tin of soup is cylindrical in shape. A cylinder
consists of a curved surface and two flat surfaces.
The flat surfaces are shaped like a circle.
You can draw circles using your pair of compasses.
The point at which you position the metal point of
your pair of compasses is the centre of the circle.
diam
s
eter
iu
rad
centre
The radius of the circle is the distance from the centre
to any point on the circle.
The diameter runs from one side of the circle through
the centre to the other side.
figure 1.39 A circle with its centre, radius and diameter.
Space
21
47 a Copy figure 1.40.
b Draw the circle that has its centre at A and
a radius of 3 cm.
c Draw the circle that has its centre at B and
a radius of 4 cm.
d Draw the circle that has its centre at C and
a diameter of 4 cm.
C
B
A
48 a Draw a square ABCD with sides measuring
6 cm.
figure 1.40
b Draw the diagonals of the square. Place an S
at the point at which the diagonals intersect.
Point S is the intersection of the diagonals.
c Draw a circle with its centre at S which passes through point
A. If you have drawn the circle accurately, the circle will also
pass through points B, C and D.
49 a Draw a rectangle ABCD whereby
AB ⫽ 6 cm and BC ⫽ 3 cm.
b Draw four circles, each with a radius of 3 cm.
Take A, B, C and D as the centres.
c The middle of side AD is M. Draw a circle
with its centre at M which passes through A
and D. What is this circle’s radius?
50 The photo shown in figure 1.41 is of a man who
is busy laying a circular patio. Explain how he is
making circles.
figure 1.41
51 Four arcs have been drawn in the square ABCD, as shown in
figure 1.42. The red arc from B to D can be drawn by placing
the metal point of your pair of compasses in A. Copy the whole
drawing. Now colour it in.
D
C
M
K
L
4 cm
4 cm
A
figure 1.42
22
N
Chapter 1
B
figure 1.43
52 Figure 1.43 on the previous page shows the square KLMN
which has been drawn using four half circles. Copy this drawing.
53 Figure 1.44 shows a circle that has been drawn using six arcs.
Copy the drawing.
4 cm
4 cm
figure 1.44
figure 1.45
54 The drawing in figure 1.45 consists of four equal circles which
fit into a single larger circle. Copy the drawing.
55 Make a nice shape using your pair of compasses.
A 56
Figure 1.46 shows a stack of tins of soup. The top view has also
been drawn.
view from above
figure 1.46
a How many tins are there in this stack?
b How many tins would you have to add to make the stack one
layer higher?
c How many tins are there in a stack consisting of six layers?
d Joris has 250 tins of equal size. He uses the same technique to
make as high a stack as possible. How many layers does his
stack have? How many tins are left over?
Space
23
1.6 Summary
Lines of vision
A line of vision extends from your eye to
the object you can see. You can use lines of
vision to indicate the limits of the area you
can see. In figure 1.47, Marja can see the
green-coloured area.
MARJA'S VIEWING ANGLE
visio
n
p. 4
line
§1.2
Space figures
p. 7
The most common space figures are shown
below.
The cylinder, cone and sphere have curved
surfaces. The other four space figures have
only flat faces.
of v
p. 10
Cubes and squares
Figure 1.49 shows a cube and its net.
The net consists of 6 squares.
A cube has 8 vertices, 12 edges and 6 faces.
The vertices can be indicated using capital
letters. In figure 1.49 BC is an edge and
BCGF a face.
figure 1.47
H
E
G
F
D
A
C
B
figure 1.49
24
Chapter 1
parked lorry
Marja
figure 1.48
§1.3
ision
line
of
§1.1
§1.4
Rectangle
p. 17
The net of a cuboid consists of six rectangles.
figure 1.50 A cuboid and its net.
Figure 1.51 shows the rectangle PQRS with the diagonal
PR. A rectangle is a square with four right angles.
A square is a quadrilateral with four equal sides.
S
R
nal
go
dia
P
Q
figure 1.51
Circles and cylinders
A cylinder has one curved face and two flat faces.
The flat faces are shaped like a circle.
In the circle shown in figure 1.52 M is the centre, AB the
diameter and MC the radius.
C
ius
p. 21
rad
§1.5
A
M
diameter
B
figure 1.52
figure 1.53 A cylinder.
Space
25
Diagnostic test
This diagnostic test can be replaced by
.
the diagnostic test on the
§1.1
1
revision exercise 1
Refer to figure 1.54.
Roderick is on a golf course.
From point R, looking between two
bushes, he can see golf balls lying on the
grass.
a How many golf balls can Roderick see?
Roderick walks along the path in the
direction of the golf balls.
b Mark a point S at which he can see
exactly 12 golf balls.
figure 1.54
2
Three objects are drawn below.
revision exercises Write down which space figures each
2, 3 object is made up of.
§1.2
a
b
c
figure 1.55
§1.3
3
Which of the following shapes are nets of a cube?
revision exercise 4
a
figure 1.56
26
Chapter 1
b
c
d
e
D
4
revision exercise 5
5
revision exercise 5
Look at the structure in figure 1.57.
a How many faces does the structure have?
b And how many edges?
c And how many vertices?
Figure 1.58 shows the beginning of a drawing of the cube
ABCD EFGH.
a Copy this figure and complete the cube.
b How many edges meet at each vertex of the cube?
c In which faces is edge CD?
d How many right angles does the face EFGH have in
reality?
figure 1.57
F
D
6
The blocks in figure 1.59 are cubes with edges measuring
revision exercise 6 1 cm.
a How many blocks is the structure made up of?
b The structure has to become a cube with edges
measuring 4 cm. How many blocks have to be added?
§1.4
7
revision exercise 7
§1.5
8
revision exercise 8
Look at the cuboid in figure 1.60.
a Which edges are 4 cm in length?
b How many edges are 2 cm in length?
c Which faces are square shaped?
d An ant walks along the edges from point H to point B.
It takes the shortest possible route. How many cm does
the ant walk?
e Draw a net of the cuboid.
f A cube is a special sort of cuboid. What is so special
about it?
a Draw a rectangle ABCD whereby AB ⫽ 8 cm and
BC ⫽ 6 cm.
b Draw the diagonals. Mark the intersection point with
an M.
c Draw the circle that has its centre at M and a radius
of 5 cm.
d How many vertices of the rectangle are on the circle?
e What is the circle’s diameter in cm?
B
A
figure 1.58
figure 1.59
H
E
G
F
4 cm
D
C
2 cm
A
2 cm
B
figure 1.60
Space
27
Revision
§1.1
1
Gym teacher De Zwaan has placed a number of
cones in the gym which are to be used for an exercise.
Three boys are looking into the gym from the changing room.
a How many cones can Ton see? First draw the lines of vision.
b How many cones can Nico and Harm see together?
c Some of the cones cannot be seen by any of the boys.
Colour these cones green.
figure 1.61
§1.2
§1.3
2
a
b
c
d
3
a Which two space figures can you see in the
object shown alongside?
b How many curved faces does this object have?
4
Which of the figures shown below is NOT the
figure 1.62
net of a cube?
If you cannot see whether a figure is the net of a
cube you have to draw it, cut it out and fold it up.
Name three space figures which have a curved face.
Name four space figures which do not have a curved face.
Which space figure do you know that has five vertices?
Which space figures do you know that have five faces?
a
figure 1.63
28
Chapter 1
b
c
R
5
Figure 1.64 shows the beginning of a drawing of the cube
ABCD EFGH.
a Copy this figure and complete the cube. Remember that you
have to mark three edges with dotted lines. Do not forget to
add the capital letters.
b Which edges meet in vertex F ?
c In which two faces is edge AB?
d Which edges have been drawn shorter than they really are?
G
F
A
B
figure 1.64
6
The structure shown in figure 1.65 is constructed from
cube-shaped blocks with edges measuring 1 cm.
a Count the blocks systematically and fill in the numbers
of blocks below.
number of blocks
lowest layer
..........
middle layer
..........
upper layer
..........
total
..........
+
figure 1.65
b How many blocks would you have to add to the structure to
make a single large cube with an edge measuring 4 cm?
§1.4
§1.5
7
8
Study the cuboid shown alongside.
a How many edges are 2 cm in length?
b Face ABCD is a square. Which face is the
same size as the face ABCD?
c Face ABFE is a rectangle.
Which faces are the same size as face ABFE ?
d How many faces are there in total?
e Draw a net of the cuboid.
H
G
E
A
a Draw the rectangle ABCD with
AB ⫽ 5 cm and BC ⫽ 3 cm.
b Draw the diagonal AC. Also draw the
figure 1.66
other diagonal.
c The intersection of the diagonals is M.
Draw a circle with its centre at M which
passes through point A.
d How many vertices of the rectangle are on the circle?
e Measure the diameter of the circle in mm.
F
3 cm
D
2 cm
C
2 cm
B
Space
29
Extra material
Super cube
A super cube consists of eight smaller cubes which
are linked in a special way. By unfolding and then
refolding the super cube you can continuously change
the pattern of colours on the faces.
You are now going to make a super cube yourself.
To do so you will use the cut-outs in the workbook.
Work on this assignment in pairs. Each of you will
work with one cut-out. Make an agreement as to
which colours you are going to use. Make sure that
the same shapes are coloured in the same colour.
Proceed as follows.
figure 1.67
Colour the cut-outs.
•
• Score the lines with a pair of scissors or the point
of your compass.
• Cut out the cut-outs carefully. Each cut-out will have a piece that needs
to be removed. Do so by simply cutting through from the outside.
Each cut-out will consist of four nets of cubes. These nets are joined at
the dotted lines.
• Fold along the lines.
• Glue the adhesive edges and put the cut-out together. Start with
gluing edge 1, then 2, etc.
For each of you, this will produce a row of four linked cubes. Refer to
figure 1.68. Use the two rectangles to stick the two rows of cubes together.
figure 1.68
30
Chapter 1
E
• Cut out the rectangles carefully.
• Join the two rows of cubes together. Stick one
rectangle to the two As and the other rectangle to
the two Bs. Make sure that the arrows are pointing
towards each other. Refer to figure 1.69.
You have now completed your super cube. Check that
you can unfold and fold up the super cube in a number figure 1.69
of different ways.
• What is the maximum number of colours that you can see on the six
faces of a super cube?
• Fold the cube in such a way that the sides with the four small green
circles, the sides with the large circles and the sides with the flowers
are on the outside.
Can you also fold the cube in such a way that none of these sides
are visible?
figure 1.70 A super cube can be used cleverly to store and display a lot of information.
Space
31
chapter
02
Numbers
There is a soft drinks machine in the school hall. If it is working, and if you
insert enough money, you can buy a can.
• What coins do we use?
You are only allowed to insert coins of 5, 20 or 50 cents in the machine.
One can of cola costs € 0.75.
• Vera has 5 and 20-cent coins. She has twenty of each. Which coins can
Vera insert into the machine to get her can of cola?
Give three possible answers.
• You want to buy a can of cola. Work out how many possible combinations
of coins you could insert into the machine.
32
Chapter 2
what you can learn
what you need
– There are different types of numbers, for instance
decimal numbers, fractions, even numbers and
prime numbers. Perhaps you know of even more.
– How to use your calculator for calculations with
numbers.
– That you first have to read the sum carefully to
know which keys you need to use.
– How to round off the result your calculator has
given you.
– How to use ratios for solving problems.
– A calculator that can also work with fractions
– Workbook and cd-rom
ICT
– Learning: digital learning line on pages 38 and 39.
– Learning: digital learning line on page 48 with the
applet Broken Calculator.
paragraph.
2.1 Arithmetic
O
1
What is cheaper, 5 exercise books for € 3 or 8 exercise books for € 5?
Why do you think so?
exercise book for
exercise book for
figure 2.1
Theory A
Arithmetic is not something you only do at school. Think about the
following problems.
– When do you have to leave home to arrive at school on time?
– How many weeks’ pocket money would you need to save to buy a
Gameboy?
For some calculations, you need to multiply.
Another word for multiplication is product.
The product of 3 and 8 is 3 ⫻ 8 ⫽ 24.
3 and 8 are called the factors of the product.
3 ⫻ 8 is 8 ⫹ 8 ⫹ 8.
Division is related to multiplication.
24 ⬊ 3 ⫽ 8, because 3 ⫻ 8 ⫽ 24.
The quotient of 24 and 3 is 24 ⬊ 3 ⫽ 8.
The sum of 8 and 11 is 8 ⫹ 11 ⫽ 19.
8 and 11 are called the terms of the sum.
2
3
34
Write the following as a product and work out
the answer.
a 8⫹8⫹8⫹8
b 7⫹7⫹7⫹7⫹7⫹7
c 0⫹0⫹0
a Which multiplication is related to the division
165 ⬊ 15?
b Calculate the quotient of 56 and 14.
c Calculate the product of the factors 11 and 8.
d Calculate the sum of 17 and 0.
e Calculate the product of 17 and 0.
Chapter 2
product
3 × 8 = 24
factors
quotient
24 : 3 = 8
because 3 x 8 = 24
sum
8 +11 =19
terms
4
This cross-number puzzle also appears in your
Fill it in.
.
across
b the sum of 12 and 7
c the product of 12 and 10
e the result of 1,000 ⫺ 445
g the quotient of 600 and 10
down
a the product of 17 and 3
b the result of 187 ⫺ 82
d the quotient of 1,000 and 4
f the product of 8 and 7
A
5
For the following questions, first write down the multiplication
or division needed before you give the answer.
a A school has 8 first-year classes of 29 pupils each.
How many first-year pupils are there in total?
b The Visser family is going camping for two weeks.
The camping fee is € 18 a day.
How much will the Vissers have to pay?
c The consumption of Mr Visser’s car is 1 to 15, which means
that he can drive 15 km on 1 litre of petrol.
During the holiday, he will drive 1,800 km.
How many litres of petrol will he need?
6
For the following questions, first write down the
multiplication or division needed before you
give the answer.
Be careful, there is superfluous information.
a An office building consists of 12 floors with
35 offices each.
The caretaker has to replace 5 light bulbs on
each floor every week.
How many bulbs does he replace per year?
b Peter has a holiday job for 4 weeks.
He works 30 hours a week and earns € 6.00
per hour.
How much does Peter earn in a week?
c An English teacher buys a number of
schoolbooks for his class costing € 7.50 each.
Each book has 60 pages. He has to pay a total
sum of € 225.
How many books has he bought?
a
c
b
d
e
f
g
figure 2.2
Numbers
35
A
7
The 174 first-year pupils of a secondary school in Deelen are
going on an excursion together with 6 chaperones.
The following remarks are about that trip.
I ‘The buses each have 45 seats.’
II ‘Altogether, the pupils are paying € 3,480.’
III ‘The dining tables seat 6 people.’
a What can you calculate using remark I? Work it out, and also
write down the division that applies to it.
b What can you calculate using remark II? Work it out.
c What can you calculate using remark III? Work it out.
Theory B
Multiplication has priority over addition.
Therefore, 2 ⫹ 7 ⫻ 5 results in 2 ⫹ 35 ⫽ 37.
I thought 2 + 7 × 5 was 45.
Multiplication or division have priority over
addition or subtraction
7⫹5 ⫻ 8⫽
7 ⫹ 40 ⫽ 47
3 ⫹ 5 ⫺ 48 ⬊ 16 ⫽
3⫹5⫺ 3 ⫽
8 ⫺ 3 ⫽5
When adding and subtracting, you work from left to right
When multiplying and dividing, you work from left to right
6⬊2 ⫻ 3⫽
3 ⫻ 3⫽9
6 ⫻ 2⬊3⫽
12 ⬊ 3 ⫽ 4
Always work out the calculation between brackets first
100 ⫺ 共7 ⫹ 8兲 ⫻ 5 ⫽
100 ⫺ 15 ⫻ 5 ⫽
100 ⫺
75
⫽ 25
36
Chapter 2
共28 ⫺ 10兲 ⬊ 9 ⫹ 2 ⫽
18
⬊9⫹2⫽
2
⫹2⫽4
That's because
you added up first.
Order of calculations
1 work out the bracketed calculation
2 multiplication and division from
left to right
3 addition and subtraction from left
to right
8
9
Copy and fill in the gaps.
b 共9 ⫹ 3兲 ⫻ 7 ⫺ 80 ⫽
a 9⫹6 ⫻ 5⫽
9 ⫹ ... ⫽ ...
. . . ⫻ 7 ⫺ 80 ⫽
...
⫺ 80 ⫽ . . .
Calculate.
a 共9 ⫹ 6兲 ⫻ 5
b 9 ⫻ 6⫹5
c 9 ⫻ 共6 ⫹ 5兲
d 8⫹3 ⫻ 7⫹2
e 8 ⫹ 3 ⫻ 共7 ⫹ 2兲
f 共8 ⫹ 3兲 ⫻ 共7 ⫹ 2兲
c 20 ⫺ 2 ⫻ 8 ⫺ 4 ⫽
20 ⫺ . . . ⫺ 4 ⫽ . . .
g 20 ⫺ 2 ⫻ 共8 ⫺ 4兲
h 20 ⫺ 2 ⫻ 8 ⫹ 4
i 共20 ⫺ 2兲 ⫻ 8 ⫹ 4
10 Bram buys 8 bottles of soft drink.
He calculates that he will have to pay
8 ⫻ 1.50 ⫹ 0.25 Euros.
a What is the total of Bram’s calculation?
b What mistake has he made?
A 11
Mr Brouwer hires a delivery van for six days,
during which he drives 450 km.
He receives the bill, which is also to be found in
. Fill in the bill.
your
figure 2.3
figure 2.4
Numbers
37
2.2 Decimal numbers
Use the
in the workbook.
O 12
The figure on the right represents the number
1.38.
a Explain this.
to
b Make drawings in your
represent
1.18
0.65
1.30
0.8
c You know that 65 is larger than 8.
Is 0.65 also larger than 0.8? Why?
figure 2.5
Theory A
The pot of Vaseline on the right holds 370 ml.
The number 370 is an integer.
The price is € 3.45.
The number 3.45 is a decimal number.
There are two numbers behind the decimal
point, therefore 3.45 is a number with two
decimals.
The number lines display a number with three
decimals.
Integers and decimal numbers can be marked on
a number line. The further right you go along
the number line, the larger the number is.
figure 2.6
0,5
0
2,8
1
2
3
9,1
4
5
66
You can see that
6.38 is larger than 6.3
6.38 is less than 6.4.
figure 2.7
38
Chapter 2
The number line shows the integers from 0 to 11.
7
8
9
10
11
The number 832.485 consists of six numerals. The position of a
numeral indicates its value.
eight hundred
The value of the first 8 is 8 ⫻ 100 ⫽ 800.
The value of the 3 is
3 ⫻ 10 ⫽ 30.
thirty
The value of the 5 is
5 ⫻ 0.001 ⫽ 0.005. five thousandths
8 3 2 . 4 8 5
hundreds
tens
single numbers
tenths
hundredths
thousandths
The number 4.82 is larger than 4.7899 because 4.82 ⫽ 4.8200,
and you know that 4.8200 is larger than 4.7899.
You can add ze
ros to a
decimal numbe
r, therefore
4,5 = 4,50
4,5 = 4,500
18,32 = 18,320
00
13 Write down the value of the numerals underlined in the
following amounts.
c 71.937
a 514.18
b 514.18
d 0.0252
14 a Draw a number line containing the numbers 0 to 6.
b Mark the following decimal numbers in their correct
positions with a pencil stroke.
1.5
0.8
3.9
4.75
0.3
3.25
15 Write down the following numbers in their correct order, starting
with the smallest.
1.105 0.9 0.906
1.01
1
0.94
1.008
1.098
0.0975
16 Write down three numbers between
a 1.01 and 1.02
b 0.98 and 0.99
c 0.599 and 0.6
d 0.25 and 0.251
17 Which number is exactly midway between
a 5.3 and 5.8
b 0.06 and 0.09
c 0.06 and 0.9
d 0.853 and 1.3
18 Rob says that there are one hundred numbers between 5.3 and 5.4.
But Bart says there are more than a thousand.
What do you think?
19 Three girlfriends go to a Bløf concert. The tickets cost € 40 each.
Tram tickets and food cost € 24.60 altogether.
How much does each girl have to pay?
Numbers
39
A 20
On Wednesday morning the kilometre counter on Manon’s
bicycle reads 371.8. On Tuesday she cycled 12.4 km, and 11.7
km on Monday. On Wednesday, she cycles 10.5 km.
a How many kilometres did the counter read by Wednesday
evening?
b What distance did the counter register on Monday morning?
Theory B
It is easy to multiply by 10, or 100, or 1,000 in your
head.
When you multiply by 1,000, you move the decimal
point 3 places to the right.
Therefore, 3.58 ⫻ 1,000 ⫽ 3,580.
When you divide by 100, the decimal point moves 2
places to the left.
Therefore, 1.4 ⬊ 100 ⫽ 0.014.
× 100
two zeros, therefore the decimal
point moves two places to the right.
: 10 000
four zeros, therefore the decimal
point moves four places to the left.
21 Calculate the following:
a 6.481 ⫻ 100
b 9,600 ⬊ 1,000
c 0.2 ⫻ 10,000
d 0.2 ⬊ 10,000
e 100,000 ⫻ 0.0052
f 7,310 ⬊ 100,000
22 a A notepad with 100 pages costs € 1.45.
How much does one page cost?
b A pack of paper containing 1,000 sheets costs € 24.90.
How many cents does one sheet cost?
A 23
Mr Gideonse takes a business trip to Japan. He takes
100,000 Japanese Yen with him, for which he paid € 915.
What is 1 Yen worth?
Theory C
Measuring the speed of a runner is very precise these
days. One well-known runner, Douglas, clocked
10.627 seconds for the 100-metre distance. But the
scoreboard showed 10.63.
His time was rounded off to two decimal places.
There are rules for rounding off.
If you round off 10.627 to one decimal place, the
result will be 10.6.
In fact, you have rounded down.
If you round off 10.627 to two decimal places, the
result will be 10.63, because you have rounded up.
If you were to round off 10.627 to an integer, the
answer would be 11.
40
Chapter 2
When rounding
off to one
decimal place,
you must write
one decimal nu
mber, therefor
e
8,03 becomes
8,0
rounding off to three decimal places?
look at the fourth decimal number
5.162278
is it less than 5?
then the third one
doesn't change
answer
5.162
rounding off to one decimal place?
look at the second decimal number
5.162278
is it 5 or more?
then increase the
first one by 1
answer
5.2
Rounding off
When rounding off to two decimals, look at the third
decimal number.
Is the third decimal number 5 or more? Round up.
Is the third decimal number less than 5? Round down.
24 a Round 8.86 off to one decimal place.
b Round 12.341 off to two decimal places.
c Round 7.653685 off to three decimal places.
d Round 123.498 off to the integer.
25 Round the number 35.46528 off to
a three decimal places
b two decimal places
c one decimal place
d the integer
b two decimal places
c one decimal place
d the integer
26 Round 8.9595 off to
a three decimal places
A 27
A signpost says Vlissingen 11
a What has the number been rounded off to?
b Which of the following might be the real
distance?
11,450 meters
11,080 meters
10,450 meters
10,948 meters
28 You can also round off to hundreds,
figure 2.8
thousands, . . .
If you round off 83,251 to thousands,
you get 83,000.
If you round 83,251 off to hundreds,
you get 83,300.
Round 8,257,139.9 off to
a millions
b thousands
c tens
d the integer
Numbers
41
2.3 Fractions
O 29
You can come across fractions in all sorts of situations: in the
street, the shops and the newspapers.
a Does 1 kg of cooking apples cost more or less than € 2.00?
b What fraction of parents does not help first-year students with
their maths homework?
c Last week, 22 football goals were scored.
How many were scored this week?
Theory A
One in 8 of the pizza slices shown on the right have
anchovy on them.
One in 8 is expressed as 18 .
3 slices have mushrooms on them.
numerator
3
For 3 out of 8, you write –
8
3
8 is a fraction.
denominator
The number above the fraction line is called the
numerator, because it counts the number of items you
have. The number below it is called the denominator,
figure 2.9 The pizza has been divided into
because each item is called one eighth.
eight slices of equal size.
2 of the 8 slices have tomato and cheese on them.
The fraction is therefore 28 .
You could also say:
1 in 4 of the slices have tomato and cheese on them.
The fraction is therefore 14 .
Therefore,
2
8
⫽ 14 .
The fraction 28 has been reduced to 14 .
42
Chapter 2
figure 2.10 As you can see,
2
8
⫽ 14 .
Similarly,
15
20
⫽ 34
both numerator and denominator can be
divided by 5
36
60
⫽ 35
both numerator and denominator can be
divided by 12
36 can also be reduced
60
step by step.
36 = 18 = 9 = 3
60
30
15
5
Reduction
You can divide the numerator and
denominator of a fraction by the same
number.
⫽ 12
5
If you separate the integer from the fraction 12
5 , you
2
can then write it as 2 5 , but you don’t have to do this.
24
10
Agreements
1. You should reduce fractions as far as possible.
2. You don’t have to separate the integer from the fraction.
30 Write down the fraction applying to the coloured sections of
each of the figures below.
a
b
c
d
figure 2.11
31 Reduce the following fractions.
a
15
25
c
18
36
e
20
100
g
35
90
i
75
100
b
15
27
d
28
35
f
56
40
h
60
12
j
32
32
or
2
5
32 a Why is
1
5
larger than 16 ? Explain it with a story.
7
10
b Which is larger?
or
5
8
7
9
or
1
2
3
10
33 a Draw a number line from 0 to 6.
b Insert the following numbers in their correct positions.
Mark them with a pencil stroke.
5 12
1
4
4
5
1 34
5 13
4 35
1
5
Numbers
43
O 34
The following addition can be applied to figure 2.12a:
2
1
3
5⫹5⫽5
a
b
c
figure 2.12
a Which addition can you apply to figure 2.12 b?
b Which addition can you apply to figure 2.12c?
Theory B
Sometimes fractions have the same denominators.
Therefore, when you add or subtract them, the denominator doesn’t
change.
e.g.: 27 ⫹ 37 ⫽ 57 en 89 ⫺ 59 ⫽ 39 ⫽ 13
reduction of the fraction
The fractions 14 and 15 don’t have the same denominator.
If you want to add them, you first have to make all the denominators
the same.
1
4
5
4
9
⫽ 20
⫹ 15 ⫽ 20
⫹ 20
The new denominator 20 was achieved by multiplying the
denominators 4 and 5.
e.g.:
2
3
9
19
⫹ 35 ⫽ 10
15 ⫹ 15 ⫽ 15
12
13
2 12 ⫺ 1 15 ⫽ 52 ⫺ 65 ⫽ 25
10 ⫺ 10 ⫽ 10
For
3
10
The denominator 30 will also work.
8
9
25
5
⫽ 30
⫹ 15
⫹ 16
30 ⫽ 30 ⫽ 6
reduction
44
the new denominator is 2 ⫻ 5 ⫽ 10
8
you don’t have to make the new
⫹ 15
denominator 10 ⫻ 15 ⫽ 150.
3
10
the new denominator is 3 ⫻ 5 ⫽ 15
Chapter 2
table
5 times
1
le
b
a
t
10 times
15
30
10
new
45
0
2
tor 0
a
n
i
6
30 enom
d
75
40
0
5
35 Work out the following:
a
1
2
⫹ 14
b
1
2
⫹ 13
c
3
4
⫺ 13
You can work these
out mentally.
d 1 12 ⫺ 14
e 2 13 ⫹ 1 14
f
3
10
1
2
1
1
2 – 3 = 13
3 + 2 5 = 55
⫺ 15
g 4 ⫺ 1 23
3
h 5 10
⫹ 1 14
A 36
Work out the following:
a
3
7
⫹ 58
5
b 1 58 ⫹ 12
c
1
36
d
7
8
1
⫹ 72
8
⫹ 64
A 37
At the Huygens College, all the
first-year pupils had a maths exam.
One third of the pupils scored 6 marks,
two fifths scored 7, and a quarter
scored 8. The rest had even higher
marks.
Which portion of the pupils scored
9 or 10 marks?
O 38
In figure 2.13a, 34 of the rectangle is red.
In figure 2.13b, 12 of the red part is blue.
a
b
figure 2.13
a Which portion of the rectangle is blue?
b How much is 12 ⫻ 34 ?
Numbers
45
Theory C
In exercise 38 you saw that
1
2
⫻
3
4
⫽ 38 .
3
⫽ 20
and
5
7
⫻
2
9
⫽ 10
63 .
Therefore,
⫻
1
4
3
5
As you can see, when multiplying fractions you must multiply both
the numerators and the denominators.
Multiplying fractions
fraction × fraction =
numerator × numerator
denominator × denominator
3× 2 =6
7 7
You can do this
without an extra
calculation.
5
With 1 14 ⫻ 23 you get 1 14 ⫻ 23 ⫽ 54 ⫻ 23 ⫽ 10
12 ⫽ 6 .
With 3 ⫻ 27 you get 3 ⫻ 27 ⫽ 31 ⫻ 27 ⫽ 67 .
a
5
8
⫻
3
7
b
1
4
⫻
5
6
c
2
9
⫻
d 1 ⫻
1
3
e 3 ⫻ 29
7
5
7
5
f 1 ⫻1
1
4
2
5
g
1
5
⫻ 15
h
3
5
⫻ 15
d
1
4
⫻ 10
2
3
a
41
1
5
b
⫻ 45
3
4
⫻ 80
c
9
10
part of 80
is
1
5
× 80 =
1
5
×
80
1
=
80
5
= 16
a The rake normally costs € 24.00. How
much will you pay at the Garden Centre?
b In the sale, a lawnmower only costs € 30.00.
What does it normally cost?
46
Chapter 2
⫻ 2000
a 14 part of 60
b half of 34
c one quarter of
d one third of 67
9
10
43 Laura has done some extra exercises with
3 1 1 1
4 =4 + 4 + 4
3
1
4 =3 x 4
3
1 1
4= - 4
3 1 1
4 =2 + 4
fractions.
Look, there they are on the right.
a Do the same sort of exercises, but in such a
way that the result is always 56 .
b Work out 6 sums where the result is
7
. At least one of them must be a
always 10
multiplication.
A stadium is chock-a-block full of football fans.
Half of them support the home team.
One fifth of the audience support the visiting
team.
The other 12,000 are neutral.
How many people have come to watch the
match?
A 45
Asia is the most highly populated continent.
The number of its inhabitants is 35 of the entire
world population.
China has 14 of the world population and India
has 15 .
a What portion of the world’s population lives
in China and India?
b What portion of the world’s population lives
in the rest of Asia?
a l
U r tains
n
Mou
A 44
A
s
i
a
China
India
figure 2.14
Numbers
47
2.4 The calculator
Use the
in the workbook.
O 46
Below is a picture of Kim van Dalen’s Post Office account
statement. Three amounts are missing. Calculate those amounts.
GIRO Account
Date
Giro Account No.
Page No.
Serial No.
1 of 1
Booked on
Name/Description
Code
Total added in Euros
Previous balance in Euros
Total deducted in Euros
New balance in Euros
No.
Giro/Bank Acc.
SUBSCRIPTION FEE TENNIS CLUB
SUBSCRIPTION YES
CLOTHES & POCKET MONEY SEPT.
TERWECHSEL LAANWEG
09:148235470 0529704
Ded.
Ded.
Add.
Ded.
figure 2.15
Theory A
Complex calculations are done with a calculator.
When entering decimal numbers, you have to use the
decimal point [.].
Check the following calculations.
3.81 ⫻ 5.2 ⫽ 19.812
Enter [3.81][*][5.2][=]
8.3 ⫺ 共2.5 ⫹ 1.7兲 ⫽ 4.1
Enter [8.3][−][(][2.5][+][1.7][)][=]
7.81 ⫺ 65 ⬊ 13 ⫽ 2.81
Enter [7.18][−][65][:][13][=]
figure 2.16
You may use your calculator for this section.
48
Chapter 2
Deductions/Additions
Amount
a 5.3 ⫻ 8.71
b 256 ⬊ 1.6
c 18.3 ⫺ 共7.2 ⫹ 2.8兲
d 8,375 ⫹ 72,830
e 32.8 ⫻ 共0.3 ⫹ 7.8兲
f 341 ⬊ 25 ⫻ 4
48 a Calculate 118 ⫺ 共51 ⫹ 16兲 and 118 ⫺ 51 ⫹ 16.
b The results of the two calculations in question a are different.
Explain why.
49 Dennis calculates 1,332 ⫻ 74 on his calculator.
The answer he gets is 18.
a How can you tell that the answer is wrong without using your
calculator?
b Dennis pressed one wrong key when entering the calculation.
Which key was it?
50 Henk has to calculate 89 ⫻ 61.
He enters [89][:][*][61][=].
a What answer will he see on the screen?
What does it mean?
b Correct Henk’s mistake. What is the answer?
51 a Calculate your age in minutes.
b How many seconds does this year still have
left?
c Janneke is 32,851,376 seconds old.
Could Janneke be a pupil at your school?
Theory B
You can do difficult calculations containing fractions
with your calculator.
Use the [a] key.
3
4
is entered like this: [3][a][4]
1 is entered like this: [1][a][3][a][4]
3
4
In the fraction 72 , the numerator is larger than the
denominator. The calculator separates out the integer
when you press [=].
After entering [7][a][2][=], the answer you will
get will be 3 12 .
The calculator can also reduce fractions:
[5][a][6][a][8][=] gives the answer 5 34 .
i.e. 5 68 ⫽ 5 34 .
figure 2.17
Numbers
49
a
2
5
⫹ 3 12
e 1 12 ⫻ 3 25
9
b 3 45 ⫺ 2 10
c
5
6
⫺ 34
d 3 ⫻
3
5
A 53
f 100 ⬊ 34
g
2
7
⬊5
h 5 ⬊ 27
2
3
1
2
Calculate the following:
a 3 12 ⫹ 2 34
c 4 12 ⫻ 2 12
b 5 34 ⫺ 78
d
3
4
⬊6
54 Arnaud is going to bake miniature pancakes for his friends.
The recipe says that he needs ¾ of a litre of milk for the mixture.
3
of a litre of milk.
However, Arnaud makes a mistake and adds 10
How much more milk should he have added?
A 55
Melco is a fruit yoghurt factory. The yoghurt is sold in 34 litre
cups. In one day, Melco makes 240 litres of fruit yoghurt.
How many cups is that?
O 56
During a collection, 37 people collected € 2,747.25.
The treasurer has calculated that the average contribution per
collector was € 742.50.
a How can you tell that this average is too high without using
your calculator?
b What mistake did the treasurer probably make while entering
the calculation?
Did I enter this correctly?
cl
ic
k!
Always check the top
line carefully.
50
Chapter 2
Theory C
It is easy to make a mistake when entering numbers in
a calculator. That is why it is important to check
afterwards whether the answer seems realistic. You
can check by estimating what the answer should be.
Estimating is done
without a calculator.
For example, the answer to 291.22 ⬊ 28.7 is
approximately 300 ⬊ 30 ⫽ 10. However, if your
calculator says 101.47, you will know for certain that
you made a wrong entry.
57 On the right, you can see part of an examination
paper.
a Without using your calculator, estimate which
of the answers are wrong.
b For each mistake, write down what went
wrong.
58 a Estimate the following answers:
286.6 ⬊ 31.5
286.6 ⫻ 31.5
11.56 ⬊ 1.89
11.56 ⫻ 1.89
50.72 ⫻ 5.1
50.72 ⬊ 5.1
b Use your calculator to check whether your
estimates were correct.
figure 2.18
59 Answer the following questions without using your calculator:
a
b
c
d
Is
Is
Is
Is
24.91 ⫻ 3.8
51 ⫻ 8.2
999 ⬊ 106
61.2 ⬊ 28.7
more or less then 100?
more or less then
2?
60 Annelies sees a beautiful pullover for € 79.50 and a scarf for € 9.95.
She has three 20 Euro notes, four 5 Euro notes, five 1 Euro
coins, and 3 other coins in small change.
Can she afford to buy the pullover and the scarf?
61 a Think of a way to estimate the number of hours you will need
to read a particular book.
b Think of a way to estimate the thickness of one page of this book.
A 62
An aeroplane flying at an average speed of 900 km per hour
reaches New York approximately seven hours after taking off
from Paris.
The Concorde is a supersonic aeroplane, with a speed of about
2,000 km per hour.
Estimate how long the Concorde’s trip from Paris to New York takes.
Numbers
51
2.5 Arithmetic in everyday situations
O 63
The pupils of class 1E are going rowing. Only 4
people fit into one rowing boat.
Roderik says: 29 ⬊ 4 ⫽ 7.25, which rounds off
to 7, so we need 7 boats.
Do you agree with Roderik? Why?
You are again
allowed to use
your calculator
in this section.
Theory A
The rules for rounding off can’t always be applied, as you have just
seen in exercise 63. Below is another example where you have to be
careful with rounding off.
Be careful with rounding off
Jeroen has a wooden plank measuring 3.60 metres.
How many planks of 65 cm can he cut out of it?
The calculation is 360 ⬊ 65 ⫽ 5.5384. . .
You must not round up to 6, because there is not enough wood
for 6 planks. The answer is 5.
64 Floor has € 2.90 in her purse.
How many packets of salted liquorice can she buy if each packet
costs 75 cents?
65 23 people are waiting for the lift on the ground floor.
7 people can fit into the lift.
How many times will the lift have to go up to bring all the
people to their destinations?
52
Chapter 2
66 A can of paint is enough to cover 3.6 m 2.
Carolien wants to paint 15 m 2.
How many cans will she need?
67 There are 219 first-year pupils at the Huygens College.
They are going on a coach tour with eight chaperones.
Each coach has 55 seats.
How many coaches will they need?
A 68
One day, 258 first-year pupils and 12 chaperones go to an
amusement park by coach. Each coach can seat 52 passengers.
The amusement park is offering a special deal: one free mobile
telephone is made available for every 25 visitors.
The pupils and the chaperones first go bob-sleighing.
Each bob-sleigh can seat 13 people, and they set off at intervals
of 45 seconds.
Write down your calculation for each of the following questions.
a How many coaches will they need?
b How many mobile telephones will they be given?
c How long does the last bob-sleigh passenger have to wait
before setting off?
69 Below you can see how four calculations have been rounded off.
For each example, think of a story that fits the calculation.
figure 2.19
O 70
a Read the newspaper article to the right.
b How many millions of m 3 of water were used
for domestic purposes?
c A household pays 85 eurocents per m 3 of
water. How many millions of Euros did all the
households pay for water in 2001?
Water in the Netherlands
In 2001, an astounding 1,674 million m 3 of
water were pumped up in the Netherlands.
525 million m 3 were used by industry,
410 million m 3 by agriculture, and the rest
was for domestic purposes.
figure 2.20
Numbers
53
Theory B
Written in full, 1,674 million ⫽ 1,674,000,000.
In the newspaper article, the word million is used
because you could easily lose count with all the zeros.
When you calculate 15 x 12.5 billion on your
calculator, enter [15][*][12.5][=].
The calculator shows 187.5. The answer is therefore
187.5 billion.
You don’t have to enter all the zeros for a billion.
It would just be a nuisance.
Large numbers
a thousand ⫽ 1,000
3 zeros
a million ⫽ 1,000,000
6 zeros
a billion ⫽ 1,000,000,000
9 zeros
300 thousand ⫽ 300,000
4.5 million ⫽ 4,500,000
23.6 billion ⫽ 23,600,000,000
71 Write the following as a number:
a thirty thousand
b 35 million
c 122 billion
A 72
At the Godafoss waterfall in Iceland, the water drops down
10 metres. 150 m 3 of water cascade down per second.
a Calculate how many m 3 of water fall in an hour.
b How many minutes will it take for a million m 3 of water
to fall?
c How many days will it be before a billion m 3 have fallen?
figure 2.21
54
d half a million
e 0.9 billion
f 150 million
Chapter 2
The Godafoss waterfall in Iceland.
A 73
a Read the newspaper article on your right.
b How do you think the newspaper reporter
arrived at the figure of ‘more than a billion
bicycles’?
c How did the reporter calculate that the
Netherlands has approximately one bicycle to
every inhabitant?
d Use the facts given in the article to calculate
how many inhabitants China has.
e Germany has 81 million inhabitants. How
many bicycles are there in Germany?
f The United States has 257 million inhabitants.
On average, how many Americans have to
share one bicycle?
The world now has more than a
billion bicycles
From our reporter
AMSTERDAM
The number of bicycles in the entire world
increased to one billion last year. China
clearly leads the list of countries, with 450
million bicycles. In the United States there
are 99 million, and in Japan 77 million. In the
whole of Europe, there are approximately
200 million bicycles riding around.
In the Netherlands we have 16 million
bicycles, which puts us in tenth place on the
listing by country. If you look at the number
of inhabitants per bicycle however, the
Netherlands wins with one bicycle per
inhabitant. Although China has by far the
largest number of bicycles, there is only one
bicycle to every 2.6 Chinese. In Germany, an
average of 1.3 people have to share a bike.
figure 2.22
A 74
Read the newspaper article about advertising.
a How many billions of Euros were spent on
advertising in the Netherlands in 1999?
b How many billions of Euros were spent on
TV commercials in 2000?
c How much was expected to be spent on
newspaper advertising in 2001?
d How many Euros were spent per inhabitant
on advertising in the Netherlands?
Advertising reaches record levels
In the year 2000, more was again spent on
advertising in the Netherlands than the year
before. Expenditure rose by € 350 million to
€ 3.45 billion. In 2000, a quarter of this
amount was spent on newspaper advertisements, and two fifths on TV commercials.
It is expected that one fifth less will be spent
on newspaper advertising in 2001 than in
2000.
figure 2.23
A 75
Weather satellites are constantly photographing
the Earth. You can see these photographs every
evening during the weather forecast.
The weather satellites are situated at an altitude
of 36,000 kilometres, and move very fast. In
one day, they travel about 270,000 kilometres.
This makes it seem as if they are in a fixed
position above the Earth.
Calculate how many kilometres a weather
satellite travels in one second.
Numbers
55
2.6 Ratios
O 76
Classes B1A and B1B are having a class evening. Esther and
Sharla will provide the crisps. They have estimated that they will
need 3 packets of crisps for every 5 pupils, but they don’t know
how many pupils will turn up.
They have therefore drawn up a table.
pupils
5
10
20
25
30
40
45
50
packets of crisps
a Copy the table and fill it in.
b The table stops at 50 pupils. Why do you think Esther and
Sharla went no further than 50?
c Ramon is supplying the peanuts. For every 5 pupils he has
estimated 2 packets of peanuts. Draw up a table for the
peanuts, similar to the one Esther made.
In the end, 45 pupils turned up.
d How many packets of crisps and peanuts were
needed?
e 32 bottles of soft drinks were bought. On the
right, you can see the price list. How many
Euros were spent altogether?
f Each pupil has to pay € 3.50. Is this enough to figure 2.24
pay for everything?
Theory A
For 5 pupils you need 3 packets of crisps, and for 20 you need
12 packets. In both cases, each pupil will get the same amount.
figure 2.25
In both cases, the ratio of the number of pupils and packets of crisps is
the same.
56
Chapter 2
You can draw up a ratio table to explain ratios.
When you multiply the numbers on the top line, you
have to multiply the numbers on the bottom line by
the same amount. You can also divide the numbers on
the top and bottom lines by the same amount.
The vertically listed numbers in the table all have the
same ratio. As you can see, the ratio 35 ⬊ 21 is the
same as the ratio 5 ⬊ 3.
This is expressed as: five to three.
×7
×4
×2
;7
pupils
5
20
35
70
10
packets of crisps
3
12
21
42
6
×4
×2
;7
×7
The ratio of the number of pupils ⬊ the number of packets of crisps ⫽ 5 : 3
Reduction of ratios
When a ratio is asked for, you should reduce it as much as
possible.
77 a Copy this ratio table and fill in the missing parts.
number of kms
90
number of minutes cycled
225
18
6
2
12
10
b Using the table, determine the ratio of 90 to 225.
78 Write down the ratios of the following:
a 12 and 24
b 27 and 18
c 75 and 15
d 32 and 40
e 30 and 54
f 7 12 and 2 12
79 Which of the following tables are ratio tables?
TABLE 1
TABLE 3
2
4
8
14
28
42
4
6
10
18
36
56
TABLE 2
TABLE 4
6
30
54
221
17
51
10
50
90
247
19
57
80 Market stall owner Albert Heldoorn sells oranges.
He is asking € 3.50 for 12 oranges.
A customer wants to buy 30 oranges.
To work out how much they cost, Albert makes a ratio table.
a Make a ratio table for Albert.
b How much do 30 oranges cost?
Numbers
57
81 Corinne is going to make doughnuts.
For a certain number of doughnuts, she needs
800 grams of flour and 300 grams of raisins.
She has enough raisins, but there are only
600 grams of flour in the house.
As Corinne wants to use all the flour, she has
to work out the correct ratio.
That’s why she makes a ratio table.
flour
800
400
200
600
raisins
a Fill in the rest of the ratio table.
b How many grams of raisins is Corinne going
to use?
A 82
Use the information below to answer the
following questions.
Maggi Mashed Potatoes is prepared from quality Dutch
potatoes. For every kilogram of mashed potatoes,
Maggi uses 6 kilograms of fresh potatoes.
The potatoes are carefully cultivated and checked.
This is necessary to achieve the best potatoes.
Maggi Mashed Potatoes can be used with a variety of
dishes, for instance fish, bubble-and-squeak, and
casseroles.
Preparation per packet:
1. Fill a pot with 400 ml water, 300 ml milk and one
teaspoon of salt, and bring to the boil.
2. Remove the pot from the stove, pour in the potato
powder and stir vigorously with a whisk or mixer.
3. Finally, stir in a small pat of butter, and your lovely
mashed potatoes are ready.
1 packet of 135 grams
is enough for 4 portions
figure 2.26
a Jolanda has read the instructions about how to prepare the
mashed potatoes. What ratio of water to milk does she need?
b Jolanda has to make enough mashed potatoes for 15 people.
Are two packets of Maggi Mashed Potato enough?
The first four lines on the Maggi packet describe how the
mashed potatoes are prepared in the factory.
c In the factory, what is the ratio of fresh potatoes : mashed potatoes?
d In the Maggi factory, it takes one hour to make 100 kg of
mashed potato powder. How many kgs of potatoes are used?
58
Chapter 2
Theory B
In many situations you will want to compare prices.
You can do this using ratio tables.
Example
At the HYPER-Market, 20 mandarins cost € 2.25.
At SUPER, the same mandarins cost € 2.60 for 24.
Which shop is cheaper?
Solution
HYPER
quantity
price
SUPER
×6
20
120
2.25
13.50
quantity
price
×5
24
120
2.60
13.00
×6
Make two ratio tables,
and work towards
reaching the same
amounts.
×5
The mandarins are cheaper at SUPER.
83 At the market, you can buy kiwi fruit at various stalls.
Bart Snelders sells 10 kiwis for € 3.25
Gerrit de Raaf sells 12 kiwis for € 3.75
Using the method shown in the example above, work out
whose kiwis are cheaper.
84 Look at the advertisement on the right.
Which of the two packages offers the cheaper
paper clips?
figure 2.27
Numbers
59
2.7 Summary
§2.1
Multiplying and dividing
p. 34
4 ⫻ 3 stands for 3 ⫹ 3 ⫹ 3 ⫹ 3.
4 ⫻ 3 is the product of 4 and 3.
In the product of 4 ⫻ 3, 4 and 3 are the
factors.
Every division is related to a multiplication.
28 ⬊ 7 ⫽ 4 because 7 ⫻ 4 ⫽ 28.
28 ⬊ 7 is called the quotient of 28 and 7.
In the sum 8 ⫹ 11, 8 and 11 are the terms.
Sequence of calculations
p. 36
1
2
3
§2.2
p. 38
ct
a produ t
n
4 x 3 is
a quotie
is
7
:
8
2
a sum
8 + 11 is
work out the calculation between brackets
multiply and divide from left to right
add and subtract from left to right
Decimal numbers
Numbers such as 0, 8 and 121 are integers.
Numbers with a decimal point, such as 3.7
and 4.21 are decimal numbers.
On a number line, you can see that 4.5 is
larger than 3.7.
When you divide by 1,000, you must move
the decimal point three places to the left.
Therefore, 3.4 ⬊ 1,000 ⫽ 0.0034
(19 − 11) : 4 + 5 =
8
:4+5=
2 +5=7
When you multiply by 1000,
you must move the decimal
point three places to the
right.
Therefor 1000 × 3.4 = 3400
Rounding off
p. 40
11.4356 rounded off to three decimal points equals 11.436.
11.4356 rounded off to two decimal points equals 11.44.
11.4356 rounded off to the integer equals 11.
§2.3
Reducing fractions
p. 42
80
12
⫽ 20
3
divide numerator and denominator by 4
28
10
⫽ 14
5
divide numerator and denominator by 2
round up
round up
round down
denominator
8
11
numerator
Adding and subtracting fractions
fractions with the a common denominator
7
4
3
1
15 ⫺ 15 ⫽ 15 ⫽ 5
when adding or subtracting the
denominator stays the same
60
Chapter 2
fractions with different denominators
3
2
9
10
19
5 ⫹ 3 ⫽ 15 ⫹ 15 ⫽ 15
first make the denominators all the same
1
36
17
19
2 ⫺ 1 16
⫽ 94 ⫺ 17
16 ⫽ 16 ⫺ 16 ⫽ 16
1
4
p. 46
numerator × numerator
denominator × denominator
Multiplying fractions
4
1 13 ⫻ 35 ⫽ 43 ⫻ 35 ⫽ 12
15 ⫽ 5
§2.4
p. 48
2
3
⫻ 60 ⫽ 23 ⫻
1
3
18
of 18 is 13 ⫻ 18 ⫽ 13 ⫻ 18
1 ⫽ 3 ⫽6
60
1
⫽ 120
3 ⫽ 40
The calculator
Complex calculations are done with a calculator.
12.3 ⫺ 共2.5 ⫹ 5.7兲 ⫽ 4.1 Enter [12.3][-][(][2.5][+][5.7][)][=]
3
5 17
Enter [5][a][3][a][17][+][11][a][5][a][7][=]
⫹ 11 57 ⫽ 16 106
119
Estimating
When you use your calculator, you need to estimate whether
the answer is correct.
An estimation of 共7.21 ⫺ 2.5兲 ⫻ 3.2 is approximately the
same as 共7 ⫺ 2兲 ⫻ 3 ⫽ 5 ⫻ 3 ⫽ 15.
§2.5
Arithmetic in everyday situations
Be careful when rounding off
p. 52
29 ⬊ 6 ⫽ 4.833. . . therefore, you can form 4 teams of 6
people from a class of 29 pupils. Do not round 4.833 up to
5, because one of the teams will then be incomplete.
p. 54
Large numbers
1 million ⫽ 1,000,000
1 billion ⫽ 1,000,000,000
1 billion ⫽ 1,000 million
8.58 million ⫽ 8,580,000
0.8 billion ⫽ 800,000,000
1,500 million ⫽ 1.5 billion
You can learn a lot from using numbers in
newspaper articles. From the article on the
right, you can see that the number of
chickens per farm is
99,000,000 ⬊ 4,400 ⫽ 22,500
§2.6
p. 56
Chickens
In 1999, the number of chickens in the
Netherlands increased to 99 million. This is
an increase of 6 million compared to 1998.
In 1999, there were 4,400 chicken farms.
Ratios
Below, you can see a ratio table.
figure 2.28
;3
×100
;5
;80
weight in kg
15
5
1
100
1.25
price in Euros
12
4
0.8
80
1
;3
;5
×100
;80
In a ratio table, you can divide and multiply the top and
bottom lines by the same amounts.
A ratio must be reduced as much as possible.
Therefore, 15 ⬊ 12 should be reduced to 5 ⬊ 4.
Numbers
61
Diagnostic test
This diagnostic test can be replaced by the diagnostic
.
test on the
§2.1
1
revision exercise 1
2
revision exercise 2
3
revision exercise 3
You are only allowed to use
your calculator for § 2.4
and § 2.5.
a Calculate the product of 8 and 6.
b Calculate the quotient of 63 and 9.
c Which multiplication relates to the
division 156 ⬊ 12?
d Fill in the relevant words: terms or
factors.
8 and 12 are . . . of 8 ⫻ 12
8 and 12 are . . . of 8 ⫹ 12
For these questions, first write down a multiplication or a division.
a Margot buys 4 kgs of bananas at the market. One kg has 7 bananas.
The price of 1 kg of bananas is € 1.75. How many bananas has Margot bought?
b Ilse sells candles at the bazaar. In one day, she sold 864 candles, which were packed
in boxes of 18. She sold € 132 worth of candles. How much does one box cost?
Work out the following, and write down all your calculation steps:
c 18 ⫹ 71 ⫺ 75 ⬊ 15 ⫻ 共11 ⫺ 7兲
a 21 ⫹ 3 ⫻ 5 ⫹ 11 ⬊ 11
d 75 ⫺ 共21 ⫹ 3兲 ⫻ 5 ⬊ 12
b 80 ⬊ 2 ⫻ 4 ⫺ 7 ⫺ 2
4
Put the following numbers in the correct order, beginning with the smallest.
revision exercise 4 8.003 8.013 7.989 7.99 8 7.0985 7.799 8.01
§2.2
5
revision exercise 5
6
revision exercise 6
7
revision exercise 7
Which number is exactly midway between
a 0.05 and 0.8
b 0.752 and 1.2
c 8.999 and 9
Calculate:
a 10,000 ⫻ 0.083
c 8.23 ⬊ 100
a Round off 13.75 to one decimal place.
b Round off 295.4497 to two decimal places.
9
revision exercise 9
62
c Round off 295.4497 to three decimal places.
d Round off 295.45 to the integer.
8
Calculate:
revision exercise 8 a 89 ⫺ 59
§2.3
b 48,500 ⬊ 1,000
Calculate:
5
a 23 ⫻ 12
Chapter 2
b
2
5
3
⫹ 10
b 1 15 ⫻ 57
c 4 15 ⫺ 2 13
d
5
12
c 2 23 ⫻ 3 13
d
5
6
5
⫺ 18
⫻ 42
D
10 Six hundred people can be seated in a hall. The entry price
revision exercise
10
is € 8.00 per person, and half-price for children. The hall is
occupied for 23 by adults, and for 14 by children.
a How many adults are present?
b How many empty seats are there?
c How many Euros has the entire audience paid?
From now on, you may use your calculator
§2.4
11 Calculate:
a 共820.46 ⫺ 498.1兲 ⬊ 2
11 b 共534.2 ⫹ 6.53兲 ⫻ 5.3
revision exercise
c 4 38 ⫹ 1 34
d 6 12 ⫺ 35
e 4 13 ⫻ 2 14
f 48 ⬊ 3 15
12 In a factory, apples are packed into sacks of 2 12 kg.
revision exercise
12
In one hour, a filling machine can fill 1200 kg of apples into sacks.
How many sacks does the machine fill per hour?
13 Estimate the answers to the following calculations. Write down how you did it.
revision exercise
a 293.5 ⫻ 21.8
b 139.8 ⬊ 19.9
13
§2.5
14 Mr Smit wants to tile his kitchen. He needs 182 tiles.
revision exercise
14
The tiles are packed in boxes of 25.
How many boxes must he buy?
15 Read the newspaper article about apples.
revision exercise
15
§2.6
a
b
c
d
How many kg was the apple harvest in 1998?
How many hectares of apple orchards were there in 1999?
Do you agree with the heading of the article? Why?
How many thousand kg of apples did an apple farmer
harvest on average from one hectare in 1999?
16 At the market, you can buy kiwis from various vegetable
revision exercises
More apples
from fewer
apple orchards
In 1999, there were 750
hectares fewer apple orchards
than in 1998, when there
were 13,500 hectares.
Nevertheless, a lot of fruit
was harvested in 1999. The
apple harvest increased by
60 million kg, to 570 million.
stalls. Whose kiwis are the cheapest?
16, 17
figure 2.29
figure 2.30
Numbers
63
Revision
§2.1
7 ⫻ 6 is the product of the factors 7 and 6. Calculate the product.
20 ⬊ 4 is the quotient of 20 and 4. Calculate the quotient.
5 ⫹ 21 is the sum of the terms 5 and 21. Calculate the sum.
Calculate the product of 11 and 7.
Which multiplication relates to the division 180 ⬊ 6?
1
a
b
c
d
e
2
A year has 52 weeks. Cindy is 832 weeks old.
You are to work out what her age is.
a Do you have to multiply or divide?
b Make the calculation. What is Cindy’s age?
Cindy earns € 30 a week.
From this you can work out how much she earns in a year.
c Do you have to multiply or divide?
d Make the calculation. What does Cindy earn in a year?
3
Order of calculating
1 first work out the calculation between brackets
2 then multiply or divide from left to right
3 then add or subtract from left to right
Example
2 ⫻ 25
2 ⫻ 25
50
50
⫺ 15 ⬊ 5
⫺ 15 ⬊ 5
⫺ 3
⫺
Calculate:
a 85 ⫺ 3 ⫻ 7
b 25 ⫹ 18 ⬊ 3
c 60 ⬊ 3 ⫻ 2 ⫹ 18
64
Chapter 2
⫻ 共2 ⫹ 6兲 ⫽
⫻
8
⫽
⫻
24
8
⫽
⫽ 26
brackets first
multiplication or division before subtraction
division and multiplication in sequence
d 20 ⫻ 3 ⫺ 21 ⬊ 7 ⫻ 共8 ⫺ 3兲
e 53 ⫺ 共7 ⫹ 1兲 ⫻ 4 ⬊ 16
f 共5 ⫺ 3兲 ⫻ 10 ⫺ 18 ⬊ 共11 ⫺ 2兲
R
§2.2
4
5
3.989 is smaller than 3.99. You will understand
this if you write 3.990 instead of 3.99.
Write the following numbers in the correct
order, beginning with the smallest:
10.103
10.03
9.989
9.099
10.001
9.99
9.859
Midway between 0.19 and 0.3 is 0.245. You will
understand this if you write 0.190 instead of
0.19 and write 0.300 instead of 0.3.
Which number is exactly midway between
a 0.6 and 0.95
b 0.03 and 0.9
c 0.832 and 1.6
989 is smalle
r than 990
therefore
3.989 is smal
ler than 3.990
difference = 0.110
half = 0.055
0.245
0.190
0.300
+ 0.055
figure 2.31
6
When you multiply by 1,000, move the decimal point 3 places
to the right, e.g. 0.0258 × 1,000 = 25.8.
When you divide by 1,000, move the decimal point 3 places to
the left, e.g. 83.7 : 1,000 = 0.0837.
Calculate:
a 10,000 ⫻ 0.0286
b 37,200 ⬊ 10,000
7
§2.3
8
c 0.1 ⬊ 1,000
d 0.09 ⫻ 1,000
a Round off 18.654 to one decimal place.
And once more to two decimal places.
b Round off 11.49 to the integer.
And once more to one decimal place.
c Round off 12.964 to two decimal places.
And once more to one decimal place.
When adding fractions, you first have to make
all the denominators the same.
Therefore
2
3
e 0.1 ⫻ 100,000
f 0.078 ⬊ 100
Rounding off 7.8496
to two decimal places
7.85
to one decimal place
7.8
to the integer
8
Be careful: with three decimal places it is 7.850
and not 7.85, because you have to show three
decimals.
8
3
11
⫽ 12
, and
⫹ 14 ⫽ 12
⫹ 12
1 13 ⫹ 16 ⫽ 43 ⫹ 16 ⫽ 86 ⫹ 16 ⫽ 96 ⫽ 32
Calculate:
3
a 10
⫹ 56
b
5
12
⫹ 16
reduce
c
2
5
⫺ 18
d 1 27 ⫹ 2 34
e 5 ⫺ 2 13
f
8
11
5
⫹ 22
Numbers
65
9
Multiplying fractions goes like this:
numerator ⫻ numerator
, i.e.
denominator ⫻ denominator
Calculate:
a 37 ⫻ 58
b
4
5
⫻2
1
4
3
5
⫻ 1 13 ⫽ 35 ⫻
4
3
4
⫽ 12
15 ⫽ 5 .
c 5 ⫻ 1 13
d
3
8
⫻
e
2
7
⫻ 28
f 2 ⫻ 67
4
9
1
3
10 a Read the newspaper article. You can use the following
method to calculate how many people in the Betuwe area
voted against.
3 80,000 . . .
3
⫻ 80,000 ⫽ ⫻
⫽
⫽. . .
4
4
1
...
b Of the 240 first-year pupils, 11/12 are taking part in
Sports Day.
How many first-year pupils are participating?
§2.4
11 a Enter 4 35 ⫹ 1 13 in your calculator as follows:
[4][a][3][a][5][+][1][a][1][a][3][=].
b
c
d
e
f
g
How much is 4 35 ⫹ 1 13 ?
Use your calculator.
1 23 ⫻ 1 34
1 23 ⬊ 1 19
12 12 ⬊ 2 12
4
1
5 ⫹ 14
共14,621 ⫺ 1,789兲 ⫻ 12
14,621 ⫺ 1,789 ⫻ 8
12 A manufacturer sells shoe polish in tubes of
3
40
kg.
A filling machine can fill 60 kg of shoe polish into
tubes per hour. How many tubes does the machine
fill per hour?
13 You can estimate the product of 698 ⫻ 19.6 by
figure 2.33
calculating 700 ⫻ 20. Your estimate comes to
700 ⫻ 20 ⫽ 14,000.
Estimate the answers to the following calculations,
and write down how you did it.
c 20.7 ⫻ 5.927
a 198.7 ⫻ 20.83
b 602.7 ⬊ 19.98
d 1,397 ⫺ 298
66
Chapter 2
Three quarters of
inhabitants against
Betuwe Railway Line
80,000 Betuwers have
voted.
figure 2.32
R
§2.5
14 With situation exercises, you can’t always round off according to
the rules. You must look carefully at the content of the story.
a Patricia wants to treat her class of 26 pupils to biscuits.
There are six to a box.
How many boxes does Patricia have to buy?
b Mrs Ruiter wants to make roller blinds. At the market, she
bought a piece of curtain material 13.20 metres long. For one
blind, she needs exactly 1.5 metres. How many roller blinds
can Mrs Ruiter make from the material she bought?
15 Read the newspaper article about pears.
Record pear harvest
a How many kg was the pear harvest in 2000?
b How many kg of Conference pears were
harvested in 2000?
c How many hectares of Conference pears were
there altogether in 2000?
d How many kg of pears were harvested per
hectare in 1999?
§2.6
Compared to the pear harvest of 130 million
kgs in 1999, there was a 70 million kgs
increase in the year 2000.
In 2000, two thirds of the harvested pears
were Conference pears. The area under pear
cultivation was 6,000 hectares in both years.
One hectare of Conference pears yielded a
harvest of 34,000 kgs of pears in 2000.
16 A packet of biscuits weighing 250 grams
contains 25 biscuits.
The following ratio table applies.
figure 2.34
×2
No. of biscuits
5
10
15
20
25
No. of grams
50
75
100
250
×2
a You can calculate the number of grams 50 biscuits weigh as
follows: 50 ⫽ 2 ⫻ 25, and you will then get 2 ⫻ 250 ⫽ 500 grams.
Copy the table and fill in the blanks.
b What is the ratio between the number of biscuits and the number of grams?
c How many grams do 60 biscuits weigh?
d Claire has weighed 180 grams of biscuits.
How many biscuits are there?
17 Eveline has a paper round. It takes her 40 minutes to deliver
60 newspapers. Jasper also has a paper round. It takes him
15 minutes to deliver 25 newspapers.
Who delivers the most newspapers in half an hour?
Explain you answer with ratio tables.
Numbers
67
Extra material
Hieroglyphic symbols
1
In hieroglyphic symbols you write 386 en 1,025 as follows:
386 ⫽
1,025 ⫽
a What number is
b What number is
c Write the following numbers in hieroglyphic
symbols:
67
405
2,009
3,572
Roman numerals
2
The river Nile flows through Egypt. About
4000 years ago, the Nile overflowed its banks
and flooded the fields every year. In those
days, there were officials whose task it was to
measure the land each year after the floods,
to document the precise borders of the fields.
They made their calculations with
hieroglyphic symbols. The following
symbols were used: for 1, for 10, for
100, and for 1000.
The Egyptians had no symbol for zero.
figure 2.35
We use the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to write numbers with. These are Arabic
numerals. They came to Western Europe via Spain in the Middle Ages and have been in general
use since 1600. Before that time, Roman numerals were used. You can still see examples of that
today, for instance when Roman numerals are used to indicate years or the times on a clock.
Roman numerals are capital letters
I for 1
X for 10
C for 100
V for 5
L for 50
D for 500
M for 1,000
MMLXVII therefore stands for 2,067, and DCCLXXXIII stands for 783. The number 9 is not
VIIII, but IX, meaning 10 ⫺ 1. Therefore, 14 ⫽ XIV, 400 ⫽ CD, and 900 ⫽ CM.
However, 49 ⫽ XLIX and not IL.
a Write down the following numbers in Arabic
numerals.
XXXVII
XLVI
CXXIV
MDC
MCDXCII MCMLXXVI
b Write down the following numbers in Roman
numerals.
68
93
298
465
502
600
1,296
1,806
c Name one difference between the Roman way of
writing numbers using I, V, X, etc., and the way
we write numbers today, using the numerals 0 to 9.
d What year is inscribed on the gable in figure 2.36?
figure 2.36
e See if you can find any buildings with Roman
numerals where you live. Write them down using normal numbers.
68
Chapter 2
E
Divisors
3
The number 3 is a divisor of 24, because the division 24 ⬊ 3
results in a whole number.
1 is also a divisor of 24, because 24 ⬊ 1 is a whole number.
The largest divisor for 24 is 24 itself, because 24 ⬊ 24 ⫽ 1.
If you divide 24 by a larger number, the result will be a fraction.
a The divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. Prove it.
b Write down all the divisors of 36.
c Write down all the divisors of 60.
d Write down all the divisors of 37.
Prime numbers
4
5
The divisors of 2 are 1 and 2. Therefore, 2 has two
divisors. The divisors of 5 are 1 and 5. Therefore 5
has two divisors. The divisors of 13 are 1 and 13.
Therefore 13 has two divisors.
2, 5 and 13 are examples of prime numbers.
a Why is 9 not a prime number?
b Why is 11 a prime number?
c Which is the only even prime number?
d Do you think that 1 is a prime number? Why?
e Write down all the prime numbers smaller than 30.
f Which is the first prime number larger than 50?
g How many prime numbers are there between 190 and 200?
a number that has
exactly two divisors is
called a prime number
In this exercise, you will learn a way to quickly
identify all prime numbers smaller than, for
example, 200. This method was discovered by the
Greek mathematician Eratosthenes about 250 years
before Christ, and is called the sieve of Eratosthenes.
Proceed as follows:
a Write down all the numbers from 1 to 200 in rows
of 10.
b Cross out number 1, as it is not a prime number.
c Draw a circle around 2, because 2 is a prime
number. Then cross out all multiples of 2, e.g. 4,
6, 8, 10 . . ., because they are not prime numbers.
figure 2.38 The sieve of Eratosthenes.
d Go back to the beginning and draw a circle around
the first number after 2 that has not been crossed out. This is the number 3.
Then cross out all multiples of 3, e.g. 6, 9, 12, 15, 18, . . .
e Repeat the process. You will be left with the prime numbers 5, 7, . . .
Numbers 69
f How many prime numbers are there that are smaller than 200?
chapter
03
Locating points
Every four years, a round-the-world yacht race sets sail from Southampton.
The participants have to cover 50,000 kms in nine months.
It is important for the yachtsmen to know exactly where they are.
Read what navigator Marcel van Triest experienced:
“We knew that there must be an island close by, but because of bad
weather we couldn’t see it. Suddenly the depth-gauge jumped from 112 m
to 10 m. At the same instant, a row of black rocks loomed 100 m ahead of
us. We were lucky, we just managed to sail around them.”
• What does a navigator do on board?
• Why is there a compass on board?
• What else does a navigator use to find out the boat’s position?
70
Chapter 3
what you can learn
ICT
– How to indicate where you are, for instance in an
office building, a school, or on a chessboard.
– How to mark a point on an axis mathematically.
– That there are other numbers besides positive
numbers, namely negative numbers, such as those
you can find on a thermometer.
– How to make calculations with this new type of
number.
– Learning: digital learning line with the computer
program Coordinates on pages 78 and 79.
paragraph.
3.1 Where are you
O
1
In figure 3.1, you can see the Huygens School. This building has
three wings. On the left is the A-wing.
figure 3.1
a
b
c
d
e
f
g
h
i
j
k
l
72
Look for classroom A12. Which classroom is situated above it?
What does the 0 in classroom A02 mean?
What does the 2 in classroom A23 mean?
How many classrooms does the A-wing contain?
Look for classroom B21. What does the letter B mean?
Look for the red classroom. What number is it?
How many classrooms are there on the second floor of the
C-wing?
The building has two staircases.
Where do you think they are located?
Wouter’s first period is in classroom A12. For the second
period, he has to go to classroom B23. He takes the shortest
route. Which classrooms does he pass?
Eline walks from C23 to B13.
Which classrooms does she pass?
Antje has to get from C23 to A03.
How many classrooms does she have to pass?
How many different routes can Antje take to get from C23 to A03?
She always takes the shortest route.
Chapter 3
On the map below you can see a part of Reykjavik. Letters and
numbers are used to locate points on a map. For example, the
Loftleidir Hotel is located in square D4.
.
See also your
C
ata
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ndu
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ur
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gbra
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Vatn
sm
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45
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114
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31
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149
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Fríkirk
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18
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rarg
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r
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Se
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ur
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als
M
134
st
au
an
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Ara
3
ga
ar
Nja
Eg
ge
r ts
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ta
Skerplug.
Fo
ssa
g.
Flu
gva
llab
rau
t
2
Hö
rpu
g.
O
jórs
árg
orr
aga
.
Air Iceland
Domestic
Air Terminal
ta
Hótel
Loftleiir
4
Ein
ars
14
ne
Bau
gat. B
a
s
ug
46
Íslandsflug
Domestic
Air Terminal
an
es
C
figure 3.2
4
Air Terminal
International
Flights
D
The Hallgrimm church is the most impressive church in Iceland.
Reykjavik has two famous churches. They are the Catholic
cathedral 共5兲 and the modern Hallgrimm church 共40兲.
a In which square is the Catholic cathedral located?
b In which squares can you find the Hallgrimm church?
Ferdinand has just visited the Hallgrimm church. He cycles
down Skolavoroustigur. He takes the first road on the left until
he can go no further, and then turns right. He takes the first road
on the left and then the third road on the left. He crosses a busy
intersection, and stops at the point where the road makes a sharp
turn.
c Which building is he standing in front of?
Locating points
73
3
On your right is a picture of a chessboard.
The squares on a chessboard are identified with
a letter and a number.
A knight is standing on square c3.
The knight can be moved to various locations,
but remember to always move him one square
straight, and one square diagonally.
Therefore, the knight on c3 can be moved to b1.
a Name the other 7 squares where the knight
can be moved to from c3.
b The white knight is standing on square e8.
Which squares can he be moved to?
c Another knight can only go to two squares
from its present position. What is its present
position? Name all the possibilities.
d Later on, one of the knights is standing on
square f6. After 4 moves he is standing on h4.
Which squares did he pass to get there?
Give two possibilities.
8
7
6
5
4
3
2
1
a
b
c
d
e
f
g
h
figure 3.3
O
4
You can indicate locations in various ways, for instance with a
letter and a number, as with the chessboard or a map.
a Name another example where a location is indicated using a
letter and a number.
b Name an example where a location is indicated using one or
two numbers.
A
5
An office building has 15 floors. Figure 3.4 shows the floor plan
of the eighth floor. Seen from the stairs, office K lies two
squares to the east and four to the north. Office K is therefore
numbered 0824.
a What number would the office on the twelfth floor, directly
above L have?
b Mrs Zeldenrust works in office 0645. Diagonally below her
room a radio is playing. In which offices could the radio be?
Mr Bode has to take a letter from office 0224 to office 1014.
c Which offices does Mr Bode pass if he takes the shortest route
via the stairs?
d Which offices does Mr Bode pass if he takes the shortest route
via the lift?
L
north
O
K
lift
T
stairs
east
figure 3.4 Floor plan of the eighth
floor. The corridor is shown in grey.
74
Chapter 3
3.2 Positive and negative numbers
O
6
In figure 3.5, you can see what the temperature in a number of
cities was on 18 December.
a What was the temperature in Amsterdam and in Moscow?
b Look at the weather report below. Mark the temperatures
.
correctly on the thermometer in your
30
25
20
Lisbon
15
10
Weather Report
5
0
Prague
Barcelona
Berlin
Helsinki
16°
⫺1°
⫺14°
London
Oslo
6°
⫺7°
Amsterdam
–5
–10
–15
Moscow
–20
c Which city had the lowest temperature?
d In which cities was the temperature below zero?
figure 3.5
Theory A
On a thermometer you can see numbers above zero,
and numbers below zero.
I can recognise negative
numbers by the minus sign.
The numbers above zero are positive numbers.
20
The numbers below zero are negative numbers.
10
0
On the thermometer in figure 3.6, the numbers are
marked vertically.
The horizontal scale below also contains negative
numbers.
-10
-20
figure 3.6 On a thermometer, you can see numbers above
and below zero.
-3 21
-4,8
-5
-4
1 21
-3
-2
-1
0
1
4,8
2
3
4
5
smaller
larger
figure 3.7
Locating points
75
Positive numbers are to the right of zero.
Negative numbers are to the left of zero.
If you go right along the scale, the numbers become larger.
If you go left along the scale, the numbers become smaller.
2 is smaller than 3, but note that -5 and -18 are also smaller than 3.
everything smaller than 3
–3
–2
–1
0
1
2
3
4
5
6
7
figure 3.8 Every number left of 3 is less than 3.
The words smaller 共less兲 than can be represented by
the symbol ⬍ .
−3 < 2 means −3 is less than 2
⫺2 is to the right of ⫺5, therefore ⫺2 is larger
than ⫺5.
The words larger 共more兲 than can be represented by
the symbol ⬎ .
−2 > −5 means −2 is larger than −5
7
a Draw a scale with the numbers ⫺5 to 5.
b Add the following numbers and mark them on
the scale with a line.
1
⫺2 14
⫺3 12
⫺0.4
3.6
2 14
3
8
Aad says: ‘Zero is a positive number.’
Meryl says: ‘Zero is a negative number.’
Neither of them is right.
Explain why.
76
Chapter 3
The symbol <
points to the left,
meaning 'less than'.
9
What is the meaning of the words negative and positive in the
following sentences?
a Football club FC KNAL has a negative goal balance.
b The suggestion made by the first-year pupils was received
positively by the mentor.
c Michiel has a negative attitude.
d You will find the negatives in the top drawer of the desk.
e Boxer Bert Barends quickly regained a positive mood after his
k.o. defeat.
10 Copy and fill in either ⬍ or ⬎ :
a ⫺9 . . . ⫺8
b 20 . . . ⫺74
c ⫺81 . . . ⫺97
d 3 12 . . . 3 14
e ⫺2 12 . . . ⫺2 14
f ⫺7.5 . . . ⫺7.3
g ⫺ 13 . . . ⫺ 23
h 2.3 . . . 2.2
i 0 . . . ⫺0.1
11 Arjan says: ‘A positive number is always larger than a negative number.’
a Is he right?
b Write down why you think he is right or wrong.
The Meteorologist
THE OCCUPATION
Ruud works as a meteorologist at the Leeuwarden airbase.
Of course, the weather forecasts he makes refer specifically
to aviation.
‘For pilots, facts about wind, thunderstorms, snow or black
ice are very important,’ he says. ‘The temperature is less
important.’
Ruud received his training with the Airforce Meteorological
Team at Woensdrecht, for which he had to have a HAVO
certificate for maths and physics. ‘To become a
meteorologist, you have to be interested in the weather and
be good with a computer, because all meteorological charts
and data arrive via the computer,’ Ruud declares. ‘But as a
meteorologist, I can work anywhere in the world. I think
that’s great.’
Locating points
77
3.3 Axes
Use the
in the workbook.
O 12
Below is the map of an island.
Point O is marked in the centre of the map.
On the map, Bergen is located at point 3, 2.
This means that from O, you reach Bergen by going 3 points to
the right, and up 2 points.
hotel
Noordermeer
swimming pool
3
Bergen
2
light-house
Town Hall
1
–5
–4
cave
–3
–2
Renes
–1
O
1
–3
Meer
a To get from O to the harbour, you first go . . .
to the right and then up . . . .
The harbour is therefore at point 共. . ., . . .兲.
b The Town Hall is located at point 共. . ., . . .兲.
c What is located at point 共3, 4兲?
d Oosterhuizen is located at point 共. . ., . . .兲.
Chapter 3
3
Visschersdorp
figure 3.9 You can indicate any point on the map with two numbers.
78
2
–1
–2
supermarket
harbour
4
Oosterhuizen
5
O 13
Look at figure 3.9.
In addition to going to the right of O, you can
also go to the left of it.
If you first go 4 to the left of O and then up 2,
you will reach the light-house. The light-house
is located at point 共⫺4, 2兲.
If you first go 2 to the left of O and then down 1,
you will reach Renes, which is located at point
共⫺2, ⫺1兲.
When locating points, you always move
sideways first, and then up or down.
a To reach Noordermeer from O, go . . . left,
and . . . up. Noordermeer is therefore located
at point 共. . ., . . .兲.
b To reach Vissersdorp from O, go . . . right,
and . . . down. Vissersdorp is therefore located
at point 共. . ., . . .兲.
c What is located at point 共⫺2, ⫺3兲?
d What can you find at points 共4, 0兲 and 共0, ⫺3兲?
e Which letter marks point 共0, 0兲?
14 Look at figure 3.10.
a From O to A is . . . left and up . . . .
Point A is therefore located at 共. . ., . . .兲.
b B is at point 共. . ., . . .兲.
c C is at point 共. . ., . . .兲.
d Which letter is located at point 共⫺3, 0兲?
e E is at point 共. . ., . . .兲.
f F is at point 共. . ., . . .兲.
g G is at point 共. . ., . . .兲.
h O is at point 共. . ., . . .兲.
E
A
2
1
F
D
–2
G
–1
O
1
2
–1
–2
C
B
figure 3.10
Locating points
79
Theory A
You can use graph paper to indicate the location of a point. First, mark
the point from which you will start counting. That point is called
the origin, which is marked with a capital O.
In figure 3.11 below, you can see the point marked O. You can also
see a horizontal line and a vertical line, both with numbers along them.
This figure is called an axis. The horizontal line is called the x-axis,
and the vertical line is called the y-axis.
y-axis
3
2
1
–4
–3
–2
–1
O
1
2
3
4
5
x-axis
–1
–2
figure 3.11 An axis.
An axis
x-axis horizontal
y-axis vertical
The x-axis, the y-axis and the origin together form an axis
How to draw an axis
1 Choose the origin.
2 Draw the x-axis and the y-axis.
3 Mark O, x and y.
4 Write in the numbers.
80
Chapter 3
Instead of writing x-axis and y-axis,
I can just mark x en y on the axis.
Figure 3.12 is an axis containing the points R, S,
and T.
R is located at point 共3, 1兲.
The numbers between brackets are called coordinates.
The first number is the x-coordinate.
The second number is the y-coordinate.
y
S
T
2
R
1
Coordinates are not always integers, as the location
of T共1 12 , 2兲 shows.
–1
O
1
2
3
x
–1
Therefore, point T is not a grid point, but the points
for R共3, 1兲 and S共⫺1, 2兲 are grid points.
figure 3.12
The coordinates of a grid point are
integers.
Coordinates
S(−1, 2) means: to get from O to S, you go 1 to the left and up 2.
first number
x-coordinate
second number
y-coordinate
A(3, 1)
René Descartes
HISTORY
René Descartes (1596-1650) was a French scholar.
He also studied in the Netherlands.
He was the first person to use an axis to indicate the
location of a point. This is why intersecting straight axes
are also called Cartesian coordinates. The story goes
that the idea came to him because of a fly in his room.
He realised that the place where the fly sat on the ceiling
could always be described by measuring from a fixed
point, the origin.
If you would like to know more about Descartes, surf to
mathematics under www.digischool.nl, and select the
history of mathematics.
Locating points
81
15 a Draw an axis. Mark the numbers from ⫺5 to 5 along both
axes.
b Plot the points P共⫺3, ⫺3兲, Q共3, ⫺3兲, R共3, 3兲, S共⫺3, 3兲,
T共0, ⫺5兲, U共5, 0兲, V共0, 5兲 and W共⫺5, 0兲.
c Draw the square PQRS.
d Draw the quadrangle TUVW. Is TUVW also a square?
16 a Draw an axis. Mark the numbers from ⫺3 to 6 along both
b
c
d
e
f
axes.
Mark the point K共2, 3兲.
Point L is located 4 squares to the right of K.
Mark point L.
L is located at point 共. . ., . . .兲.
Point M is located 2 squares above K.
M is located at point 共. . ., . . .兲.
Point N is located 4 squares below K.
N is located at point 共. . ., . . .兲.
P共. . ., . . .兲 is two squares to the left of K.
17 Answer the following questions without plotting the points.
a Point 共. . ., . . .兲 is located four squares to the right of point A共3, 1兲.
b Point 共. . ., . . .兲 is located five squares to the left of point B共3, 11兲.
c Point 共. . ., . . .兲 is located three squares below point C共11, 81兲.
18 a Draw an axis and plot the following points. Then join the first
point to the second one, the second to the third, and so on.
Finally, join the last point to the first one.
共⫺3, 5兲, 共⫺8, 4兲, 共⫺3, 6兲, 共⫺2, 7兲, 共0, 7兲, 共⫺2, 2兲, 共1, 4兲,
共5, 3兲, 共7, 4兲, 共6, 0兲, 共6, ⫺1兲, 共3, ⫺1兲, 共2, ⫺2兲, 共2, ⫺3兲,
共⫺1, ⫺3兲, 共1, ⫺2兲, 共1, ⫺1兲, 共⫺1, 0兲, 共⫺3, ⫺1兲, 共⫺4, 1兲,
共⫺3, 4兲 en 共⫺6, 2兲.
b You can make a nice drawing from this and then colour it in.
19 a Draw an axis. Mark the numbers
from ⫺3 to 5 along both axes.
b Plot the points P共⫺2, 3兲 and Q共4, 3兲.
Join P and Q.
c On the segment PQ, there are 5
A segment always has a starting
grid points between P and Q.
point and an end point. A segment
Colour the grid points red, and write
does not go on forever.
down their coordinates.
82
Chapter 3
20 This exercise is only about the points A共0, 5兲, B共⫺2, 2兲,
C共⫺1, 3兲, D共1, ⫺2兲, E共⫺3, 4兲, F共4, 0兲, G共⫺2, 1兲, H共5, ⫺1兲,
K共2, ⫺3兲 and L共3, 3兲.
Answer the following questions without plotting the points.
a Which point is located at x-coordinate 5?
b Which point is located at y-coordinate -2?
c Which two points have the same x-coordinate?
d Which two points have the same y-coordinate?
e Which point is located on the x-axis?
f Which point is located on the y-axis?
21 Four triangles are shown in the figure below. They increase in
size from left to right.
y
D
C
3
B
2
1
A
1
O
1
2
2
3
3
4
5
4
6
7
x
8
figure 3.13
a Copy the table below and then fill it in.
number of triangle
highest point
1
2
3
4
5
A共1, 1兲
B共. . ., . . .兲
C共. . ., . . .兲
D共. . ., . . .兲
E共. . ., . . .兲
b Muriël has copied figure 3.13, and she has added some more
of those increasingly larger triangles.
What are the coordinates of the highest point H of her eighth
triangle?
c Geert has copied figure 3.13 onto a very large piece of paper,
and has added lots of triangles on the right-hand side.
What are the coordinates of the highest point of his 10th and
18th triangles?
d Geert can see that the coordinates of the highest point of his
39th triangle are 共77, 39兲.
What are the coordinates of the highest point of triangle 41?
Locating points
83
A 22
In figure 3.14, you can also see triangles that are progressively
larger, towards the left this time. You could even add some more
on the left-hand side.
y
D
4
C
3
2
B
A
1
4
–7
–6
3
–5
–4
–3
–2
2
–1
O
1
1
figure 3.14 These triangles increase in size towards the left.
a Make a table similar to the one in exercise 21a.
b The y-coordinate of the highest point of one of the
triangles is 9.
What is the x-coordinate of that point?
c The x-coordinate of the highest point of one of the
triangles is ⫺25.
What is its y-coordinate?
23 The points A共0, 2兲, B共4, 4兲, and C共8, 6兲 are the beginning of a
long line of points that are equally far apart.
a What are the coordinates of the fourth and fifth points along
the line?
b What are the coordinates of the eighth and tenth points?
c The coordinates of the 20th point are 共76, 40兲. What are the
coordinates of the 21st point?
d The y-coordinate of one of the points along the line is 62.
Find its x-coordinate.
A 24
84
There is a recognisable pattern in the following rows of points.
Write down the coordinates of the two points that follow each
row. Describe the regular patterns in your own words.
a 共0, 5兲
...
共2, 10兲
共4, 15兲
共6, 20兲
b 共3, 7兲
...
共6, 14兲
共9, 21兲
共12, 28兲
c 共3, 1兲
...
共5, 0兲
共7, ⫺1兲
共9, ⫺2兲
d 共⫺1, 2兲
...
共⫺6, 4兲
共⫺11, 8兲
共⫺16, 16兲
Chapter 3
2
3
x
25 a Write down the coordinates of four points along the x-axis.
b Write down the coordinates of four points along the y-axis.
c Copy the following sentences and fill in x or y.
For all points along the x-axis, the . . . coordinate is 0.
For all points along the y-axis, the . . . coordinate is 0.
Between 3 en 3?
Then 3 en 3 don't
count any more!!
If you are sure that they are right, colour them with a marker.
26 a Draw an axis.
b There are five grid points whose y-coordinate is 2, and whose
x-coordinates lie between ⫺3 and 3.
Colour these grid points red.
c Colour all grid points blue whose x-coordinate is ⫺1, and
whose y-coordinates lie between ⫺2 and 4.
d One of the points has both colours. Write down its
coordinates.
A 27
a Draw an axis.
b Colour all the grid points red whose x-coordinates lie between
⫺1 and 4, and whose y- coordinates lie between ⫺3 and 2.
A 28
In figure 3.15, you can see progressively larger
squares.
a Write down what the coordinates of the corner
points of square 5 would be.
b A corner point of one of the squares has the
coordinates 共⫺7, 10兲.
Write down the coordinates of the other corner
points of that square.
c Work out what the coordinates of square
number 50 would be.
y
5
4
3
2
1
1
2
–2
–1
O
–1
1
2
3
4
5
x
3
–2
figure 3.15
Locating points
85
3.4 Adding and subtracting positive numbers
O 29
One day, the temperature in Leeuwarden was ⫺4 degrees.
At night, it was 3 degrees colder.
What was the temperature that night?
Theory A
Below you can see a sort of scale with which you can
play a number game for two people.
It contains the integers from ⫺10 to 20.
Each player has one pawn and starts at 0.
Take it in turns to throw a dice.
If, for instance, you throw a 5, you can
either move your own pawn 5 squares forward
or move your opponent’s pawn 5 squares back.
You will win the game if
either your own pawn passes 20
or your opponent’s pawn passes -10.
But be careful! At 13 there is a well. If a pawn lands
on it, the player has to go back to ⫺4.
At ⫺7, there is an ejector seat. If a pawn lands on it, it
shoots forward to 7.
figure 3.16 The numbers game-board.
86
Chapter 3
30 a Marit and Theo are playing the game. Marit’s pawn is
standing on 3, and Theo’s is on ⫺2. It is Marit’s turn, and she
throws a 6. She can
either move her own pawn 6 places forward to . . .
or she can move Theo’s pawn back by 6 to . . .
b Theo’s pawn is standing on 2. It is Marit’s turn, and she puts
Theo’s pawn back to ⫺3. What did Marit throw?
c Play the game a few times.
31 a Fons’ pawn is standing on ⫺6. He goes forward 4 places.
Which square is he on?
b Mies’ pawn is standing on 3. She has to go back 5 places.
Which square should she move to?
c Nelli’s pawn is standing on ⫺5. She has to go back 3 places.
Which square should she move to?
32 You can make calculations with this game.
Adding
Fons is on −6 and goes forward 4 places.
He arrives at −2.
The calculation is −6 ⫹ 4 ⫽ −2
Subtracting
Nelli is on −5 and has to go back 3 places.
She arrives at −8.
The calculation is −5 ⫺ 3 ⫽ −8
a Sheila is on 5 and goes forward 4 places.
The calculation is 5 ⫹ . . . ⫽ . . .
b Pieter is on ⫺1 and goes back 3 places.
The calculation is ⫺1 ⫺ . . . ⫽ . . .
c Aleid is on 8 and goes back 2 places.
The calculation is 8 ⫺ . . . ⫽ . . .
d Jasper is on ⫺4 and goes forward 5 places.
The calculation is ⫺4 ⫹ . . . ⫽ . . .
33 Johnny plays the game. He writes down the calculation for each
move he makes with the pawn. The first part of some of his
calculations is written below. Complete his calculations.
a 6 ⫺ 1 ⫽ ...
d ⫺1 ⫹ 4 ⫽ . . .
g ⫺8 ⫹ 2 ⫽ . . .
b 2 ⫹ 5 ⫽ ...
e ⫺1 ⫺ 4 ⫽ . . .
h ⫺8 ⫺ 2 ⫽ . . .
c 2 ⫺ 5 ⫽ ...
f ⫺1 ⫹ 1 ⫽ . . .
i 0 ⫺ 6 ⫽ ...
34 On an axis, there is a point whose x-coordinate is ⫺4.
This point is moved 12 squares to the left.
a What is the new x-coordinate?
b How would you calculate this?
35 Think of a short story to fit the calculation ⫺4 ⫹ 9.
Write down your story, and complete the calculation.
Locating points
87
Theory B
You can easily indicate additions and subtractions using arrows on a
scale.
The scale below illustrates the addition ⫺3 ⫹ 5 ⫽ 2.
5
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
0
1
figure 3.17
to add 5: the arrow goes 5 places to the right
The scale below illustrates the subtraction ⫺1 ⫺ 7 with an arrow.
You start at ⫺1 and go 7 to the left, ending up at ⫺8.
The calculation is ⫺1 ⫺ 7 ⫽ ⫺8 .
7
–9
–8
–7
–6
–5
–4
–3
–2
–1
figure 3.18
to subtract 7: the arrow goes 7 places to the left
36 a Draw an arrow scale for the addition ⫺7 ⫹ 2, and work out
the result.
b Draw an arrow scale for the subtraction ⫺3 ⫺ 4, and work
out the result.
37 Calculate the following. You may use arrows, but you can also
work it out in your head.
a ⫺4 ⫹ 6
b ⫺9 ⫹ 5
c ⫺6 ⫺ 4
d ⫺5 ⫺ 9
e
f
g
h
0 ⫹ 17
⫺3 ⫹ 3
⫺15 ⫺ 25
⫺8 ⫺ 26
38 Try to calculate the following in your head.
a
b
c
d
e
f
g
h
88
6⫺8
⫺1 ⫹ 3
⫺4 ⫺ 9
⫺2 ⫹ 10
27 ⫺ 31
⫺8 ⫺ 21
12 ⫺ 10
⫺6 ⫹ 33
Chapter 3
i
j
k
l
m
n
o
p
⫺76 ⫺ 29
57 ⫺ 62
⫺13 ⫹ 23
⫺59 ⫹ 83
⫺213 ⫺ 76
⫺131 ⫹ 67
456 ⫺ 456
⫺765 ⫺ 0
4
5
39 You can also add or subtract three numbers. Do this in two steps.
Example
A 40
⫺17 ⫺ 19 ⫹ 12 ⫽
⫺36 ⫹ 12 ⫽ ⫺24
Fill in the following
a ⫺4 ⫺ 3 ⫺ 5 ⫽
... ⫺ 5 ⫽ ...
b ⫺5 ⫹ 8 ⫺ 10 ⫽
. . . ⫺ 10 ⫽ . . .
Calculate:
a ⫺6 ⫹ 9 ⫺ 5
b 3⫺7⫺1
c ⫺16 ⫹ 29 ⫺ 35
d ⫺14 ⫺ 17 ⫹ 36
e
f
g
h
⫺39 ⫹ 19 ⫹ 7
⫺12 ⫺ 3 ⫺ 29
⫺14 ⫹ 14 ⫺ 6
33 ⫹ 17 ⫺ 50
41 Mrs Van Ham has a debit of € 83 on her Giro account.
You could also say: Mrs Van Ham has € ⫺83.
She deposits € 120 into her Giro account.
a Write down the addition you will use to work our her new
balance.
b What is the balance of her Giro account now?
GIRO Account
Date
Giro Account No.
Page No.
Serial No.
of
Mrs
Total Additions in Euros
Previous Balance in Euros
Total Deductions in Euros
New Balance in Euros
DEBIT
Booked on
Name/Description
DEPOSIT OWN ACC
Code
No.
Giro/Bank Acc.
Deductions/Additions
Amount
ADD
figure 3.19
Locating points
89
42 Newspaper reports dated 20 August 2001.
Heat-wave in Budapest
Heat-wave at the South Pole
Michael Schumacher managed to win his
fourth Formula 1 World Championship under
a burning sun. With the temperature at
31 degrees Celsius, Schumacher stayed ahead
of his opponents.
This year, the South Pole is experiencing its
warmest winter in 35 years. On Sunday, the
temperature reached -7 degrees Celsius.
Normally, temperatures are considerably
lower.
By how many degrees was it warmer at the race in Budapest
than at the South Pole?
43 Professor Barabas has built a new time-machine.
You have to move through six chambers in it. In figure 3.20 you
can see what happens to time in each chamber. Several routes are
possible. Using the red route 0 ⫹ 7 ⫺ 9 ⫹ 4 ⫺ 3 ⫺ 5 ⫹ 2, you
will go back 4 years in time.
a Calculate the time difference using the blue route.
which will put you 1 year
b Draw a route in your
ahead in time.
figure 3.20
90
Chapter 3
3.5 Adding and subtracting negative numbers
O 44
Frank has two bank accounts. The one has a balance of € 64, the
other has a balance of € ⫺12. To find out his combined balance,
Frank adds up the two balances, 64 ⫹ ⫺12 ⫽ . . .
ACCOUNT
ABC • BANK
MR. F. PIETERSEN
SALDOSTRAAT 12
1234 AB AMSTERDAM
ACCOUNT
ABC • BANK
MR. F. PIETERSEN
SALDOSTRAAT 12
1234 AB AMSTERDAM
Acc. Type (in Eur.)
Account No.
Savings Acc.
12 . 34 . 56 . 789
Previous Balance
New Balance
87 , 28 + / CREDIT
64 , 00 + / CREDIT
Interest Date Description
24 - 05
CASH POINT 24 .05 . 02 / 16 . 09 HRS.
EDAH , PAS 123
TERMINAL AB1234
Personal Limit
Acc. Date
250
27 - 05 - 2002
Total Deductions
23 , 28
Amount Deducted (Debit)
23 , 28
Acc. Type (in Eur.)
Account No.
Savings Acc.
12 . 34 . 56 . 456
Previous Balance
New Balance
43 , 01 + / CREDIT
12 , 00 - / DEBIT
Interest Date Description
20 - 05
CASH POINT 20 .05 . 02 / 09 . 20 HRS.
ABC , PAS 123
No. of Pages Pages
Page 123123
Serial No
TERMINAL
1
001
21
21Total
- 05Additions CASH POINT 21 .05 . 02 / 13 . 21 HRS.
TANGO , PAS 123
TERMINAL 2RSN1234
Amount bib (credit)
Acc. Date
Personal Limit
250
23 - 05 - 2002
Total Deductions
55 , 01
Amount Deducted (Debit)
10 , 00
No. of Pages
Page No. Serial No.
1
001
21
Total Additions
Amount Added (Credit)
45 , 01
figure 3.21
Theory A
In the column you can see 15 ⫹ ⫺4 ⫽ 11
But you know
15 ⫺ 4 ⫽ 11
15 + 2 = 17
15 + 1 = 16
15 + 0 = 15
15 + −1 = 14
15 + −2 = 13
Therefore, 15 ⫹ ⫺4 gives the same result as 15 ⫺ 4.
15 + −3 = 12
15 + −4 = 11
Examples
9 ⫹ ⫺3 ⫽ 9 ⫺ 3 ⫽ 6
⫺9 ⫹ ⫺3 ⫽ ⫺9 ⫺ 3 ⫽ ⫺12
15 ⫹ ⫺16 ⫽ 15 ⫺ 16 ⫽ ⫺1
⫺15 ⫹ ⫺16 ⫽ ⫺15 ⫺ 16 ⫽ ⫺31
哭哭哭哭哭哭
Look at the column of additions on the right. Each time, 1 less is
added to 15 and, of course, the result keeps getting less as well.
1 less
1 less
1 less
1 less
1 less
1 less
+ − is the same as −
45 Calculate the following according to the
examples.
Therefore also write down the step in-between.
a 5 ⫹ ⫺1
e ⫺6 ⫹ ⫺20
b 10 ⫹ ⫺5
f 0 ⫹ ⫺8
c ⫺9 ⫹ ⫺1
g 13 ⫹ ⫺21
d ⫺5 ⫹ ⫺6
h 19 ⫹ ⫺19
Locating points
91
46 Calculate:
a ⫺8 ⫹ ⫺3
b ⫺8 ⫹ 3
c 8 ⫹ ⫺3
d ⫺8 ⫺ 3
e 8⫺3
f ⫺3 ⫺ 8
g 3 ⫹ ⫺8
h ⫺3 ⫹ ⫺8
c ⫺43 ⫹ ⫺57
d 3 ⫹ ⫺40
e 121 ⫺ 135
f ⫺141 ⫹ 97
g ⫺56 ⫹ ⫺56
h 91 ⫹ ⫺117
47 Calculate
a 14 ⫺ 23
b ⫺20 ⫹ ⫺42
A 48
Example
⫺15 ⫹ ⫺17 ⫹ 6 ⫽
⫺15 ⫺ 17 ⫹ 6 ⫽
⫺32
⫹ 6 ⫽ ⫺26
In the same way, calculate:
a 7 ⫹ ⫺5 ⫹ 10
b 7 ⫹ ⫺5 ⫹ ⫺10
c ⫺7 ⫹ ⫺5 ⫹ ⫺10
O 49
d ⫺18 ⫹ ⫺17 ⫹ ⫺16
e 25 ⫹ ⫺34 ⫺ 18
f ⫺17 ⫹ 45 ⫹ ⫺28
Frank has two bank accounts. The one has a balance of € 64, and
the other a balance of € ⫺12.
a What is the difference between the two amounts?
b You can also calculate this by subtracting the two amounts
from each other. Then you get 64 ⫺ ⫺12 ⫽ . . .
Theory B
15 − 2 = 13
15 − 1 = 14
15 − 0 = 15
15 − −1 = 16
In the column you can see 15 ⫺ ⫺4 ⫽ 19
But you know
15 ⫹ 4 ⫽ 19
Therefore, 15 ⫺ ⫺4 gives the same result as 15 ⫹ 4.
Examples
8 ⫺ ⫺3 ⫽ 8 ⫹ 3 ⫽ 11
⫺7 ⫺ ⫺5 ⫽ ⫺7 ⫹ 5 ⫽ ⫺2
0 ⫺ ⫺10 ⫽ 0 ⫹ 10 ⫽ 10
⫺13 ⫺ ⫺27 ⫽ ⫺13 ⫹ 27 ⫽ 14
50 Calculate the following according to the examples.
Therefore also write down the step in-between.
a 2 ⫺ ⫺1
e ⫺7 ⫺ ⫺ 20
b ⫺2 ⫺ ⫺1
f ⫺14 ⫺ ⫺18
c ⫺2 ⫺ ⫺6
g 26 ⫺ ⫺21
d 2 ⫺ ⫺8
h ⫺19 ⫺ ⫺19
92
Chapter 3
15 − −2 = 17
15 − −3 = 18
15 − −4 = 19
哭哭哭哭哭哭
How much is 15 ⫺ ⫺4?
You can find that out by using the column of subtractions on the right.
Each time, 1 less is subtracted and, of course, the result keeps getting
more as well.
1 more
1 more
1 more
1 more
1 more
1 more
− − is the same as +
A 51
Calculate the following. Be careful! There are both additions and
subtractions.
a ⫺7 ⫺ ⫺8
d ⫺1 ⫹ ⫺5
g 35 ⫺ ⫺16
j 16 ⫹ ⫺43
b ⫺7 ⫹ ⫺8
e 1 ⫹ ⫺5
h ⫺12 ⫹ ⫺24
k ⫺23 ⫺ 0
c 7 ⫺ ⫺8
f 1 ⫺ ⫺5
i ⫺6 ⫺ ⫺33
l 0 ⫺ ⫺23
A 52
Calculate:
a ⫺7 ⫹ ⫺7
b ⫺7 ⫺ ⫺7
c 7 ⫹ ⫺7
A 53
d ⫺18 ⫺ ⫺33
e ⫺5 ⫺ ⫺22
f 57 ⫹ ⫺75
g ⫺93 ⫹ 92
h ⫺59 ⫹ ⫺73
i ⫺54 ⫹ ⫺54
j ⫺214 ⫺ ⫺67
k ⫺214 ⫹ ⫺67
l ⫺123 ⫹ ⫺45
Below you can see a table with additions and one with
subtractions.
The red square contains the result of ⫺1 ⫹ ⫺3.
The yellow square contains the result of 3 ⫺ 7.
Copy the two tables and fill in the blanks.
⫹
⫺5
⫺3
⫺4
⫺1
3
7
⫺
⫺5
⫺3
7
⫺4
⫺4
⫺1
3
⫺4
54 You can also do additions and subtractions with negative
numbers on your calculator.
Work out how to do the calculations in exercise 52 with your
calculator.
Locating points
93
3.6 Summary
§3.1
p. 72
Locating points
In schools, office buildings, hotels and hospitals, rooms or
offices are often numbered 210, 405, and so on. Room 405
means that the room is ‘room 5 on the fourth floor’.
On a chessboard, the fields are indicated with a letter and a
number. You can also find this method of indicating locations in atlases and computer games.
§3.2
p. 75
Negative numbers are numbers below zero.
You can recognise them by the minus sign in front.
On a scale, the negative numbers are to the left of 0,
whereas the positive numbers are to the right of 0.
The number 0 is neither positive nor negative.
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
figure 3.22
⫺4 ⬍ 0
⫺4 ⬎ ⫺6
means
means
⫺4 is less than 0.
⫺4 is larger than ⫺6.
The further you go right along a scale, the larger the
numbers will be.
The further you go left along a scale, the smaller the
numbers will be.
§3.3
Axis
p. 78
On the right is an axis. An axis consists of
an x-axis and a y-axis. The x-axis is
horizontal, and the y-axis is vertical.
The point where the x-axis crosses the
y-axis is called the origin, or O.
y
3
C
–4
–3
F
–2
–1
1
figure 3.23
Chapter 3
B
1
O
2
3
–1
–2
94
A
2
D
E
x
Point A共3, 2兲 means:
in order to get from the origin O to A,
you have to move 3 to the right, and up 2.
F共⫺4, ⫺1兲 means:
In order to get from O to F, you have to
move 4 to the left, and down 1.
A(3, 2)
x-coordinate
The points A, B, D and F are grid points.
You can think of this in two ways.
– the coordinates are integers
– the points are located on the intersection
of two grid lines.
y-coordinate
For every poin
t along the
x-axis, the y-coo
rdinate is 0.
For every poin
t along the
y-axis, the x-c
oordinate is 0
.
The points C共⫺1, 12 兲 and E共2 12 , ⫺1 12 兲 are
not grid points.
The point B共1, 0兲 is located on the x-axis.
The point D共0, ⫺2兲 is located on the y-axis.
The coordinates of O are 共0, 0兲.
§3.4
Adding and subtracting positive numbers
p. 86
You can indicate an addition on a scale with an arrow
pointing to the right.
The scale below shows the addition ⫺5 ⫹ 7 ⫽ 2.
7
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
1
2
3
4
5
figure 3.24
If you subtract 5, the arrow should point to the left.
The scale below shows the subtraction ⫺1 ⫺ 5 ⫽ ⫺6.
5
–7
–6
–5
–4
–3
–2
–1
0
figure 3.25
You can also add or subtract three numbers. Make a step
in-between. ⫺17 ⫺ 9 ⫹ 5 ⫽
⫺26 ⫹ 5 ⫽ ⫺21
§3.5
p. 91
Adding and subtracting negative numbers
7 ⫹ ⫺5 ⫽ 7 ⫺ 5 ⫽ 2
⫺2 ⫹ ⫺14 ⫽ ⫺2 ⫺ 14 ⫽ ⫺16
⫺13 ⫺ ⫺8 ⫽ ⫺13 ⫹ 8 ⫽ ⫺5
15 ⫺ ⫺12 ⫽ 15 ⫹ 12 ⫽ 27
+
is
is +
Locating points
95
Diagnostic test
.
test on the
§3.1
1
The Cicero College teachers’ post boxes are situated in the
staff room.
1 The post boxes are numbered from 1 to 120. In the figure
below, you can see the first 17 numbers.
revision exercise
figure 3.26
a In the figure, you can see three post boxes marked x, y and z.
What are the numbers of those post boxes?
b The post boxes of the maths teachers are 12, 23, 26, 65,
86, 105, 107 and 109.
.
Colour those boxes red in your
The administration office has complained that the way
the post boxes are numbered is awkward.
c Why do you think administration finds this way of numbering awkward?
d Invent a way of numbering the post boxes that administration will find more convenient.
§3.2
Copy and fill in ⬍ , ⬎ or ⫽.
a ⫺3 . . . 4
c 5.6 . . . ⫺4
2, 3 b ⫺4 . . . ⫺5
d ⫺3.4 . . . ⫺3.44
2
revision exercises
§3.3
3
a Draw an axis with the points C共3, 2兲 and D共⫺2, 4兲.
b Draw a square ABCD in such a way that the points A
and B are located under the x-axis.
4
c Write down the coordinates of A and B.
revision exercise
96
e ⫺3 58 . . . ⫺3.76
f ⫺2 18 . . . ⫺2.125
Chapter 3
D
This exercise is about points A共5, ⫺4兲, B共⫺4, 3 12 兲, C共⫺2, ⫺2兲,
revision exercise D共4, 0兲, E共0, 4兲, F共4, ⫺5兲, G共2 14 , ⫺4兲 and H共⫺2, 2兲.
5 a Which point has the y-coordinate ⫺2?
b Which of these points have the same x-coordinate?
c Which point is located on the y-axis?
d Which points are not grid points?
e If you go 2 to the left and down 3 from point A, you
will arrive at point Q. Give the coordinates of Q.
4
5
The table below is about the triangles on the
right, which get progressively larger towards
6 the left. You can add some more.
y
revision exercise
E
5
triangle
number
top righthand corner
1
2
3
4
4
5
D4
3
C
3
5
B
2
2
A
1
A共5, 1兲
B
C
D
E
– 5 –4 –3 –2 –1 O
1
1
2
3
4
5
x
figure 3.27
a What are the coordinates of corner H of triangle 8?
b One of the right-hand corner points has the y-coordinate 15.
What is the x-coordinate of that point?
6
a Draw an axis.
revision exercises b Colour all grid points blue whose x-coordinate is ⫺2,
and whose y-coordinate lies between ⫺3 and 4.
7, 8
7
Calculate:
revision exercises a 8 ⫺ 12
9, 10 b ⫺5 ⫹ 7
c ⫺13 ⫹ 5
d ⫺22 ⫹ 11
§3.4
e
f
g
h
⫺721 ⫹ 0
⫺17 ⫺ 13
⫺23 ⫹ 14
⫺24 ⫺ 11
8
Calculate with the step in-between.
a 5 ⫹ ⫺3
e ⫺12 ⫺ 4 ⫹ ⫺10
11, 12, 13 b ⫺3 ⫺ ⫺1
f ⫺24 ⫺ ⫺24 ⫺ 24
c ⫺11 ⫺ ⫺3
g 41 ⫹ ⫺52 ⫺ 73
d ⫺43 ⫹ ⫺39
h ⫺71 ⫹ ⫺71 ⫺ ⫺71
§3.5
revision exercises
⫺
⫺12
4
9
⫺11
revision exercise
⫺13
Copy the table on the right, and then fill in
the missing numbers.
14 As you can see, 4 ⫺ ⫺5 ⫽ 9.
⫺5
⫺2
9
Locating points
97
Revision
§3.1
1
Figure 3.28 on the right is a map of an old
residential area. The letters and numbers of the
squares have disappeared. However, it is known
that the church is located in square D2, and the
smithy in square E3.
a In which squares are the mill and the bakery?
b What is in square A4?
village
hall
school
square
mill
smithy
village green
SMEDERIJ
church
bakery
figure 3.28
§3.2
2
a Copy the scale shown below in your exercise book, and then
circle the negative integers in red.
-5
-4
-3
-2
-1
0
1
2
3
4
5
figure 3.29
b Add the following numbers to the scale and mark them on it.
3.4
4 14 ⫺2.7 0.5 ⫺4.6 ⫺3.2 ⫺0.3
3
On the scale in figure 3.30, you can see that ⫺5
is to the left of ⫺3, therefore ⫺5 ⬍ ⫺3.
You can also see that ⫺3 12 ⬎ ⫺3 14 , because
⫺3 12 is to the right of ⫺3 14 .
Fill in ⬍ , ⬎ or ⫽ .
a ⫺3 . . . ⫺2
b 0 . . . ⫺2
c ⫺2.4 . . . ⫺2.2
d ⫺6 12 . . . ⫺6 14
§3.3
4
98
-3 34 -3 12
-6
-5
figure 3.30
e
f
g
h
0 ... 1
⫺3.5 . . . ⫺3 12
⫺3.7 . . . ⫺3.9
⫺ 23 . . . ⫺0.66
a Draw an axis with the points K共⫺1, ⫺3兲, L共3, ⫺1兲
and M共2, 1兲.
b Draw the segments KL and LM.
Complete the quadrangle KLMN.
c Write down the coordinates of N.
d Draw the diagonals of the quadrangle. Mark the letter T
where they cross. Write down the coordinates of T.
Chapter 3
-4
-3
-2
R
5
a Look at figure 3.31.
A is located at 共⫺2, 1兲 means:
In order to reach A from O, you first have
to go . . . left, and then up . . . .
b The x-coordinate of point A is . . ., and the
y-coordinate is . . . .
c To get from O to B共⫺1, ⫺2兲, you first
have to go . . . left, and then down . . . .
d Give the coordinates of C.
e Why is point C not a grid point?
f Write down the coordinates of points D
and E.
y
E
2
A
–3
–2
1
–1
O
1
–1
B
–2
2
3
x
C
D
–3
figure 3.31
§3.4
6
There is a regular pattern in the following rows
of points. Write down the coordinates of the two
next points in each row.
a 共0, 7兲, 共1, 14兲, 共2, 21兲, 共3, 28兲, . . .
b 共3, 5兲, 共4, 7兲, 共6, 10兲, 共9, 14兲, . . .
c 共5, 1兲, 共2, 3兲, 共⫺1, 5兲, 共⫺4, 7兲, . . .
7
a Draw an axis.
b There are four grid points whose y-coordinate
is ⫺1, and whose x-coordinates lie between
⫺4 and 1. Colour these points red.
c Colour the five grid points green whose
x-coordinate is ⫺2, and whose y-coordinates
lie between ⫺3 and 3.
d One of the points has both colours.
Write down the coordinates of that point.
Between 4 and 1,
means 4 and 1
are excluded.
8
Draw an axis and colour all the grid points blue
whose x-coordinates lie between ⫺3 and 2, and
whose y-coordinates lie between ⫺1 and 2.
9
You may use a scale to help you with the following questions.
When adding, the arrow points to the right, and when
subtracting, the arrow points to the left.
Calculate:
a 5 ⫺ 10
c ⫺3 ⫺ 5
e ⫺4 ⫺ 12
g ⫺7 ⫹ 12
b ⫺5 ⫹ 4
d 7 ⫺ 12
f ⫺7 ⫺ 12
h ⫺12 ⫺ 11
Locating points
99
10 Calculate the following, working from left to right.
18 ⫺ 7 ⫺ 14 ⫽
11 ⫺ 14 ⫽ ⫺3
⫺13 ⫹ 5 ⫺ 7
e 12 ⫺ 15 ⫺ 33
22 ⫺ 24 ⫹ 5
f ⫺14 ⫺ 21 ⫹ 8
⫺11 ⫺ 11 ⫹ 3
g ⫺23 ⫹ 15 ⫺ 8
12 ⫺ 7 ⫹ 17
h ⫺11 ⫹ 11 ⫺ 11
Example:
a
b
c
d
§3.5
11 When adding a negative number, you must first subtract.
Therefore: 7 ⫹ ⫺4 ⫽ 7 ⫺ 4 ⫽ 3.
Calculate the following; first write down the step in-between.
a 8 ⫹ ⫺4
e ⫺13 ⫹ ⫺3
b 7 ⫹ ⫺6
f ⫺34 ⫹ ⫺11
c 12 ⫹ ⫺11
g ⫺23 ⫹ ⫺12
d 24 ⫹ ⫺7
h 0 ⫹ ⫺5
12 When subtracting a negative number, you must first add.
Therefore: 8 ⫺ ⫺3 ⫽ 8 ⫹ 3 ⫽ 11.
Calculate the following; first write down the step in-between.
a 8 ⫺ ⫺4
e ⫺13 ⫺ ⫺12
b 7 ⫺ ⫺8
f ⫺49 ⫺ ⫺11
c 4 ⫺ ⫺11
g ⫺34 ⫺ ⫺34
d 17 ⫺ ⫺13
h ⫺83 ⫺ ⫺23
⫺18 ⫺ ⫺15 ⫹ ⫺12 ⫽
⫺18 ⫹ 15 ⫺ 12 ⫽
⫺3
⫺ 12 ⫽ ⫺15
Calculate the following in the same way.
a ⫺16 ⫹ ⫺11 ⫹ 12
d ⫺45 ⫹ ⫺23 ⫺ ⫺15
b ⫺56 ⫺ ⫺23 ⫹ 14
e ⫺23 ⫹ ⫺45 ⫺ ⫺78
c 34 ⫺ ⫺23 ⫺ ⫺11
f ⫺56 ⫺ ⫺34 ⫹ ⫺23
13 Example:
14 Look at the adding and subtracting tables below.
The red square shows the result of ⫺3 ⫹ 4.
The green square shows the result of 8 ⫺ 7.
Copy the tables and fill in the remaining squares.
⫹
⫺7
⫺3
100
4
1
⫺2
⫺
⫺5
7
⫺12
0
8
⫺8
⫺7
Chapter 3
0
1
+ – is –
– – is +
Extra material
Various grids
Grids do not always consist of squares.
Figure 3.32 shows a fish on a normal grid.
The same fish has been transferred to a special grid in figure 3.33.
This transfer was done with the help of the coordinates of a number
of points. As you can see, the result is an entirely different fish.
4
3
4
3
2
2
1
O
1
2
3
4
5
figure 3.32 Fish on the normal grid
1
1
Figure 3.34 below shows three different grids. They are also
pictured in your
.
Plot the following points in each of the three grids and join the
first point to the second one, the second to the third, and so on.
共5, 1兲, 共8, 4兲, 共9, 3兲, 共9, 6兲, 共6, 6兲, 共7, 5兲, 共4, 2兲 and 共5, 1兲.
10
10
9
9
8
8
7
7
6
4
3
2
1
O
6
10
9
8
7
6
5
4
3
2
1
5
1 2 3 4 5 6 7 8 9 10
elongated grid
O
5
O
1
2
figure 3.33
3
4
5
Fish on the special grid
10
9
8
7
6
5
4
3
2
1
O
1
4
3
2
2
3
4
5
1
curved grid
6
slanted grid
7
8
9
10
figure 3.34 You can find enlarged versions of these three grids in your workbook.
Locating points
101
2
3
4
5
Plot the following points in each of the three grids in your
.
共1, 2兲, 共3, 2兲, 共3, 4兲, 共4, 4兲, 共4, 6兲, 共3, 6兲, 共3, 7兲, 共5, 7兲, 共5, 9兲,
共1, 9兲 and 共1, 2兲.
Join the first point to the second, the second to the third, etc., as
before.
10
9
8
7
Using a normal grid, draw a circle with its centre point at 共5, 5兲
and a radius of 5 cm.
.
Transfer this circle to the slanted grid in your
O
In the grid shown in figure 3.36, the squares are not of equal
size. You are going to make a drawing on this grid in your
.
y
3
2
1
2
3
4
x
figure 3.36 A drawing on this grid is called an anamorphosis.
a Plot the points 共0, 12 兲 and 共0, 3兲. Join these two points with a
thick black line.
b Do the same with points 共0, 3兲 and 共⫺1, 4兲, and points 共0, 3兲
and 共1, 4兲.
c Plot the following points and join them with a flowing, curved
line.
共0, 2 12 兲, 共⫺1, 3兲, 共⫺2, 4兲, 共⫺3, 3兲, 共⫺3, 2兲, 共⫺2 12 , 1 12 兲, 共⫺1 12 , 1 34 兲,
共⫺ 12 , 1 12 兲, 共⫺2, 1兲, 共⫺1, 0兲, 共0, 1兲, 共1, 0兲, 共2, 1兲, 共 12 , 1 12 兲,
共1 12 , 1 34 兲, 共2 12 , 1 12 兲, 共3, 2兲, 共3, 3兲, 共2, 4兲, 共1, 3兲 en 共0, 2 12 兲.
d You have now drawn an animal. Colour it in. Maybe you think
this animal is misshapen. Try looking at it from where the eye
is drawn. You will now see the animal in its normal proportions.
102
Chapter 3
5
4
3
2
1
figure 3.35 Drawing on a curved
grid.
4
1
–3 –2
–1 O
6
10
9
8
7
6
5
4
3
2
1
Figure 3.35 shows a drawing on a curved grid.
Transfer this drawing to the elongated grid in your
.
E
Quadrants
y
An axis is made up of four parts.
Each part is called a quadrant. The quadrants are
numbered with Roman numerals.
2
II
I
1
quadrant
position
I
II
III
IV
upper right
upper left
lower left
lower right
–1
O
1
2
x
–1
III
IV
figure 3.37
The x-axis and the y-axis divide the grid into
4 quadrants.
The axes are not part of the quadrants.
6
What do the terms quadrant, quarter-hour and quarter turn have in common?
7
For this exercise, use the following points: A共10, ⫺2兲, B共⫺5, 74兲,
C共0, ⫺6兲, D共⫺20, ⫺50兲, E共⫺8, 0兲, F共30, 0兲, G共0, 57兲,
H共4 15 , 5 17 兲, K共⫺ 12 , 13 兲, L共7 12 , ⫺4兲, M共1900, 1兲 and O共0, 0兲.
a Which of these points are located on the x-axis, and which are on the y-axis?
b Write down in which quadrants the other points are located.
8
9
a Draw an axis with the points A共3, 2兲 and B共5, 2兲.
b Draw the segment AB.
c Extend the segment AB a long way at both ends.
You have now drawn a line through A and B.
Through which quadrants does this line stretch?
A line can be extended
at both ends.
The line doesn’t end.
a Draw an axis with the point A共⫺3, 1兲.
b Can you draw a line through A that fits exactly
into one quadrant?
c Draw all the lines through A that fit precisely into two quadrants.
d Draw a blue line through A that fits precisely into three quadrants.
e Can you draw a line that fits into all four quadrants?
10 A共1, 3兲 and B共4, 2兲 are part of square ABCD. Points C and
D are not located in the first quadrant. Complete the square.
Locating points
103
Skills
Solving problems
1
Problem
Place the numbers 3, 4, 5, 6, 7 and 8 in the open circles. Do it
in such a way that the sum of the numbers on each side is 18.
Theory A
There are many exercises in an arithmetic book.
You know instantly how to do some of them, for instance
– calculate the product of 18 and 35
– calculate 8 ⫻ 共3 ⫹ 7兲
– round off 18.235 to two decimal places.
With some of the other exercises, you first have to try them out
or think about them, as with the problem above.
In order to solve this type of problem, you really need to think
carefully. Don’t just guess at something.
First ask yourself the following questions:
– What is this about?
– What facts have I got?
Once you have worked that out, look for the correct method.
In this paragraph, you will learn the method of TRIAL AND
TESTING.
System for solving problems
1
2
3
4
104
understand the question
look for the correct method
work the problem out using the method
check your answer
Chapter 3
figure 3.38
Work systematically.
Fast solutions lead
to dead ends.
The problem shown on the right can be
solved with the four-step system as follows.
Step 1 Understand the question
– You may only use each number once.
– The three numbers on each side must add
up to 12.
Problem
Place the numbers
1 to 6 in the open
circles in such a way
that the sum of the
numbers on each
side is 12.
figure 3.39
Step 2 Look for the correct method
In this case, we have chosen the TRIAL AND TESTING method.
You can start your trial in several ways.
I You could write the numbers 1, 2, 3, 4, 5 and 6 on six small
pieces of paper, and move them around until the answer is
right.
II You could place the number 1 in one of the corners, and then
see what the consequences are.
4
5
figure 3.40
1
o1
1
ad
1
o1
pt
du
pt
du
ad
up
to
pt
du
ad
10
2
0
ad
d
o1
4
Step 4 Check your answer
to
8
3
o8
ad
d
up
2
pt
du
ad
4 ⫹ 3 ⫹ 5 ⫽ 12, 4 ⫹ 2 ⫹ 6 ⫽ 12 and 5 ⫹ 1 ⫹ 6 ⫽ 12.
As you can see, the method of TRIAL AND TESTING can be
applied in several ways. Of course, you only have to solve a
problem one way.
6
1
Step 3 Work the problem out using the method
I After trying for a while, you will reach the answer.
II – Place the number 1 in one of the corners, and then see what
the consequences are.
You need two numbers twice to add up to 11, but you don’t
have them, for only 5 ⫹ 6 ⫽ 11.
– Place the number 2 in one of the corners.
This also gives rise to problems, for only 4 ⫹ 6 ⫽ 10.
– And so on. With 4 in the corner, you need two numbers
which add up to 8. This works with 2 ⫹ 6 ⫽ 8 and
3 ⫹ 5 ⫽ 8.
If you then place the 1 between the 5 and the 6, you have
solved the problem.
2
3
6
5
figure 3.41
Locating points
105
Problems
2
Place the numbers 3, 4, 5, 6, 7, and 8 in the open circles of figure
in such a way, that the sum of the
3.42 in your
numbers on each side is 15.
figure 3.42
3
Place the numbers from 1 to 9 in the open circles of figure 3.43
in your
in such a way, that the sum of the numbers
on each side is 17.
4
In figure 3.44, the number on each of the curves is the sum of
, place the
the two neighbouring circles. In your
correct numbers in the open circles.
figure 3.43
44
38
62
figure 3.44
5
Place the numbers from 1 to 19 in the open circles in your
in such a way that the sums for all the lines made
up of three circles are the same.
figure 3.45
106
Chapter 3
6
7
Place the numbers 2 to 9 in the squares of figure
3.46 in your
. You will get two
amounts consisting of four numbers each.
Do this in such a way that
a the sum is as large as possible
b the difference between the two amounts is as
small as possible.
+
figure 3.46
Place the numbers 1 to 9 in the squares of figure
3.47 in your
.
Do this in such a way that all rows, columns and
diagonals produce the same sum.
figure 3.47
8
Figure 3.48 is a multiplication consisting of
letters. Each letter represents a number.
Work out the multiplication correctly in your
.
A B C D
4
×
D C B A
figure 3.48
9
Place the numbers 1 to 9 in the open circles of
.
figure 3.49 in your
Do this in such a way that, when you follow the
arrows, the numbers increase from small to
large.
figure 3.49
Locating points
107
chapter
04
Diagrams
The photograph shows a herring catch in the North Sea.
Each year, biologists estimate how many herrings there are in the North Sea.
These estimates are used to determine how many tons of herring each country is
allowed to catch.
108
Chapter 4
HERRING POPULATION IN THE NORTH SEA
× one thousand tons
herring stock
critical level
2000
1500
1000
800
500
1998
1996
1994
1992
1990
1988
1986
1984
1982
1980
1978
1976
1974
1972
1970
1968
0
1966
• During which period did the herring
population decrease?
• How many tons of herring were there in
1982?
• The green line indicates the critical level.
What is meant by ‘critical level’?
• The average weight of a herring is
approximately 100 grams.
One ton is 1000 kg.
Calculate how many herrings were
swimming in the North Sea in 1996.
• In two periods there was a ban on fishing.
Can you identify these periods?
year
what you can learn
ICT
– What the difference is between a pictogram, a bar
chart and a graph.
– How to read graphs.
– How to convert a table into a graph.
– What a diagram is.
– How you can use diagrams as a special way of
providing information.
– Learning: digital learning line on pages 113 and
114 with the applet Diagrams.
– The computer paragraph with the NS timetable on
page 135.
paragraph.
4.1 Pictograms and bar charts
O
1
The sea lion population of the Galapagos Islands varies from
year to year. The pictogram below reflects the numbers for one
of the islands. The census date is always 1 January.
SEA LIONS ON THE GALAPAGOS ISLANDS
2002
2001
2000
1999
1998
1997
= 50 SEA LIONS
figure 4.1
a How can you tell that on 1 January 1998 there were
approximately eighty sea lions?
b Copy the table below and fill in the blanks.
year
number of sea lions
1997
1998
1999
2000
2001
2002
80
c In which year was the sea lion population at its lowest?
Medellin
gdalena
RICA
PANAMA
Carac
Orino
c
The Galapagos Islands is
VENEZUELA an archipelago of thirteen
VENEZUE
Bogota
Buenaventura
Cali
GalápagosIslands (Ec.)
Guayaquil
110
Chapter 4
COLOMBIA
Rio Ne
Quito
ECUADOR
J apurá
Iquitos
TRINIDAD EN TOBAGO
large islands and hundreds
GUYANA
of small ones, situated in
SURINAME
the Pacific
Ocean toFr.
the
Guyana
west of Ecuador.
Because of their isolated
location, the islands are
populated with animals
that can be found nowhere
else on Earth.
Theory A
The previous page showed a pictogram where the
pictures in it tell you what they are counting.
A diagram like this is easy to understand.
However, one of the disadvantages is that drawing all
those pictures is a lot of work.
It is quicker to make a bar chart from the statistics
contained in the table, as you can see in figure 4.2.
SEA LIONS ON THE GALAPAGOS ISLANDS
quantity
400
350
300
250
200
150
100
50
0
1997
1998
1999
2000
2001
2002
figure 4.2
Rules for making a bar chart.
1 Write a title at the top of the bar chart.
2 The bars must be free-standing.
3 Describe what the numbers on each axis stand for.
2
As part of the sports selection, Patricia’s heart rate has been
measured in three ways for the last five years, once sitting down,
once standing, and once after intensive exercise on the home
trainer. The data are reflected on the bar chart below.
PATRICIA'S HEART RATE
no. of heartbeats per minute
200
180
160
sitting
140
120
standing
100
after exercise
80
60
40
20
0
1998
1999
2000
2001
2002
figure 4.3
a What was the difference between her standing and sitting heart
rates in 2000?
b Each year, there was a significant difference between her heart
rate after exercise and when she was sitting.
In which year was the difference the greatest?
c Has Patricia’s condition improved or deteriorated during these
years? Explain why you think so.
Diagrams
111
3
In 2002, Volvo dealership Janssen sold the models S40, S60 and S80.
JANSSEN LTD. CAR SALES
quantity
12
10
S40
8
S60
6
4
S80
2
0
1
2
3
4
quarter
figure 4.4
a Which model sold the best in the first quarter?
b How many model S60s were sold during 2002?
c In which quarter did Mr Janssen sell the most cars?
In 2001, Mr Janssen sold 24 S40s, 20 S60s and 12 S80s. Quite by
chance, he sold the same quantity of each model in each quarter.
d Draw a bar chart that reflects these facts. Use the same colours
as in figure 4.4.
A
4
In addition to Schiphol, there are a number of other smaller
airports in the Netherlands. The bar chart below shows the
number of passengers per airport in 1999.
NUMBER OF PASSENGERS IN 1999
number × 1000
700
600
500
400
300
200
100
0
Groningen Maastricht Rotterdam Eindhoven
figure 4.5
a How many more passengers flew from/to Rotterdam than from/to Eindhoven?
b How many times more passengers did Maastricht handle than Groningen?
c In 1999, Schiphol handled more than 36 million passengers.
How many cms tall would the bar have to be to show the
number of passengers who flew from/to Schiphol?
112
Chapter 4
4.2 Rises and falls
Use the
in the workbook.
O
5
From the diagram below, you can see that the temperature at
10 a.m. was 2 °C.
TEMPERATURE CHANGES ON A COLD DAY IN NOVEMBER
°C
10
5
O
2
4
6
8
10
12
14
16
18
20
22
24
time
–5
figure 4.6
a What was the temperature at 8 a.m. and
at 8 p.m.?
b At what time was the temperature at its
highest? What was the temperature at that
time?
c What is the difference between the highest
and the lowest temperatures?
d At which times was the temperature 5 °C?
e By how many degrees did the temperature
rise between 8 a.m. and midday?
f The lowest temperature is always measured
half an hour before sunrise.
When did the sun rise?
Diagrams
113
6
Sandra and Marsja are competing in a
400-metre race.
a What is the situation after 20 seconds?
b After how many seconds has Sandra
overtaken Marsja? How many metres
have they run by then?
c Who wins the race? How many metres
has the other one still to run?
d What is the difference in seconds at the
finishing line?
400-METRE RACE
m
400
350
300
250
200
Marsja
150
Sandra
100
50
O
10
20
30
40
50
60
70
80
90 100
secs
figure 4.7
A
7
The figure below shows two vertical axes and
two graphs.
no. of CTVs
in millions
8
no. of black & white
TVs × 1000
1 600
1 400
7
1 200
6
1 000
5
800
4
600
3
400
2
200
1
0
’65
’70
’75
’80
’85
’90
The earliest experimental TV broadcasts took place in 1948.
’95
year
figure 4.8
a Which of the graphs refers to the left-hand axis? How can you tell?
b How many black & white TVs were there in 1990, and how
many CTVs?
c Were there more CTVs or more black & white TVs in 1980?
d In which year do you think the number of black & white and
colour TVs was equal?
114
Chapter 4
O
8
During a match, the temperature in the
gymnasium varied considerably, as you can see
in figure 4.9.
a When did the temperature in the gymnasium
rise?
b When did the temperature in the gymnasium
fall?
c When did the temperature remain constant?
TEMPERATURE IN THE GYMNASIUM
°C
first half
interval second half
figure 4.9
Theory A
Some graphs have no numbers marked along the horizontal and
vertical axes, although each axis has an annotation of what it
represents. Look at figure 4.9. You can’t accurately tell what the
temperatures are. The object of this graph is to show only rises and
falls, i.e. the course of the graph. Graphs like this are called
general graphs.
General graphs
A general graph is about the course of a graph.
Usually, there are no numbers along the axes.
Graphs can rise, run horizontally, or fall. These three situations can
occur in one graph, as in figure 4.10.
temperature
Read the graph
from left to right.
ris
f al
lin
g
g
in
constant
time
figure 4.10
Rising and falling
The section of the graph pointing upwards is said to be
rising.
The section of the graph running horizontally is said to be
constant.
The section of the graph pointing downwards is said to be
falling.
Diagrams
115
9
Wouter is driving to school on his scooter.
The general graph on the right tells you
something about his speed.
a Which sections of the graph are rising?
b Which sections are falling?
c Which sections are constant?
d Along the way, Wouter once had to brake
hard, and on another occasion he had to drive
around a bend. Which situation occurred first?
e Did Wouter come to a halt at any time?
WOUTER ON HIS SCOOTER
speed
e
b
i
d
c
f
h
a
g
time
figure 4.11
10 Jens has been on a bicycle trip. In figure 4.12,
you can see the general graph of the distance he
cycled.
a Jens rested for a while during his trip. How
can you tell this from the graph?
b After his break, he cycled more slowly than
before. How is this shown on the graph?
c Think of a reason why he was cycling more
slowly.
A 11
A 12
Carolien is in class 1AH2, and is cycling to
school from her home. She cycles at a steady
pace at first, until she has to wait at a railway
crossing. Once the train has passed, she quickly
pulls away and cycles to school at a high speed.
When she arrives, she brakes hard and then
parks her bicycle.
a Draw a general graph of Carolien’s trip.
Mark the horizontal axis time, and the vertical
axis speed.
b Also draw a general graph of the distance
from Carolien’s house. Mark the horizontal
axis time, and the vertical one distance.
On the right, there is a general graph of the level
of water in a rain barrel.
Think of a story that fits the graph.
JENS’ BICYCLE TRIP
distance
time
figure 4.12
WATER IN THE RAIN BARREL
level
time
figure 4.13
116
Chapter 4
4.3 Drawing graphs
O 13
A puppy called Jasper was born on 14 October. His weight has
been recorded every day since he was born. The data are shown
in the table below.
PUPPY JASPER’S GROWTH RATE
no. of days since birth
weight
0
1
2
3
4
5
6
7
8
9
300
400
420
450
500
500
620
650
700
750
You can draw a graph using the numbers from the table.
A start has been made in figure 4.14 below.
PUPPY JASPER
You can tell from the
table how long the axes
have to be.
weight in grams
400
300
200
100
O
1
2
3
time in days
figure 4.14
a How many centimetres long is the horizontal axis going to be?
How tall will the vertical axis be?
b Copy figure 4.14 and draw the axes to the correct lengths.
c You can see from the table that Jasper’s weight was 400 grams
after one day. The coordinates of that point are 共1, 400兲.
Mark that point as shown in figure 4.14.
d Plot all the other points from the table,
starting with 共0, 300兲.
e Draw the graph by linking all the points.
The graph is not a straight line, but a
flowing curve.
f On which day did Jasper gain the most
weight? How can you tell that from the
not like this,
graph?
FLOWING CURVE
but like this
Diagrams
117
Theory A
You can draw a graph from a table. This is how you do it.
I always add
a heading at
the top.
Drawing a graph from a table
1 Draw the horizontal axis using the numbers from the table’s
top row. Always mark the steps the same distance apart.
2 Draw the vertical axis. Note the highest number and
distribute the others evenly along the axis.
3 Name the axes.
4 Plot the points from the table.
5 Draw the graph by joining the points.
14 The table on the right shows the temperature of the seawater
Seawater temperature
along the Riviera during a calendar year. The temperature was
measured on the first day of every month.
a Draw a graph from this table.
b When Francien was on holiday, the sea’s temperature was
18 °C. When was Francien there on holiday?
c Carlijn says that the temperature of the sea doesn’t rise above
22 °C. Is she right? Explain why.
d At a depth of 50 metres, seawater is 3 degrees colder.
On the graph you made for question a, draw another graph in a
different colour, showing the temperature at a depth of
50 metres.
month
degrees C
January
14
February
14
March
15
April
16
May
17
June
19
July
22
August
22
September
22
October
21
15 Every morning, proud father Arjan Kruise weighs his newly-
November
16
born daughter Lotte. The table below shows the results.
December
15
LOTTE’S WEIGHT IN GRAMS
day
weight
0
1
2
3
4
5
6
7
8
9
10
3000
2960
2930
2910
2900
2920
2940
2980
3030
3080
3130
To prevent the vertical axis from becoming too long, a section of
it has been left out in figure 4.15.
You can see that by the tear line 共the zig-zag section兲.
a How tall would the diagram be if the tear line had not been
inserted?
b Copy the diagram and make the axes longer. Consult the table
to see how much longer they should be.
c Draw the graph.
d How many days did it take for Lotte to recover her weight at
birth?
118
Chapter 4
LOTTE'S WEIGHT
grams
3 100
3 000
2 900
0
1
2
3
4
day
figure 4.15 The tear line has
been inserted to prevent the
graph from becoming too tall.
O 16
The graph below shows the price of almond cookies.
PRICE OF ALMOND COOKIES IN THE CANTEEN
eurocents
80
60
40
20
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
figure 4.16
a
b
c
c
O 17
How often was the price increased in the course of these years?
What did an almond cookie cost on 12 April 2001?
Was there a year in which an almond cookie cost 45 eurocents?
This graph is not a flowing curve. It consists of a number of
horizontal lines. Why is that?
This graph shows the price of a number of bags
of crisps.
a How much do five bags of crisps cost?
b Bart has € 1.25. How many bags of crisps can
he buy?
c The graph consists of a number of floating
points. Why?
THE PRICE OF CRISPS
eurocents
250
200
150
100
50
0
1
2
3
4
5
6
no. of bags
figure 4.17
d Simon buys seven bags of crisps while the supermarket is running
a special offer. How much does he have to pay for them?
e Draw a graph representing the special offer.
Diagrams
119
Theory B
In the previous exercises, you saw three types of graph.
Which type of graph you should draw, depends on the situation.
FLOWING CURVE
HORIZONTAL LINES
FLOATING POINTS
weight
price per cookie
price of crisps
age
You can join the points,
because any weight and age
are possible.
year
quantity
You can only buy whole bags
of crisps. Therefore, do not
join these points.
The price remains the same
for a certain time.
Only then does the price
increase.
figure 4.18
18 The figure below shows three advertisements. Assign the correct
graph to each of the advertisements.
1
Everything has to go.
Carpeting at € 25 a
metre.
price in Euros
2
Dirt cheap!
Double CDs
€ 25 per box.
3
price in Euros
Catamarans for hire.
€ 25 per hour or part of
an hour.
price in Euros
100
100
100
75
75
75
50
50
50
25
25
25
1
Graph A
figure 4.19
120
Chapter 4
2
3
1
Graph B
2
3
1
Graph C
2
3
19 Work out which type of graph belongs to which situation.
Choose either I ⫽ flowing curve, II ⫽ horizontal lines, or
III ⫽ floating points.
BICYCLE TRIP
ROLLS OF CANDY
AUNT JO'S DIET
speed
price
weight
time
no. of rolls
a
time
b
c
POSTAGE FOR LETTER
PRICE OF CARTON MILK
PHOTOGRAPH PRINTS
price
price
price
weight
d
time
e
no. of photos
f
figure 4.20
A 20
The figure below shows the price-list of a parking garage.
Draw a graph using the information provided.
figure 4.21
A 21
Think of a situation where a graph consists of
a a flowing curve
b horizontal lines
c floating points.
Diagrams
121
4.4 Graphs and relationships
O 22
The screw of an outboard motor of a boat rotates
very quickly. The graph in figure 4.22 shows
how many rotations the screw makes at top
no. of rotations of screw
speed.
8.000
a How many rotations does the screw make in
6.000
two minutes?
b The graph goes through point 共4, 6,000兲.
4.000
What does this mean?
2.000
c The following points can be read along the
0
graph 共1, . . .兲, 共1 12 , . . .兲, 共. . ., 4,500兲 and
1
2
3
共. . ., 6,750兲.
figure 4.22
d The screw carries on rotating for some time.
Extend the graph in your mind’s eye.
Through which points would it pass?
共10, 15,000兲
共14, 20,000兲
共32, 48,000兲
共200, 300,000兲
e Complete the table of this graph.
no. of minutes
1
2
3
9
30
90
no. of rotations
Theory A
The following table gives you some information on
Angela’s walk.
×3
×2
×2
time in minutes
10
20
30
60
distance in metres
800
1600
2400
4800
×2
×2
×3
As you can see, the table shows relationships.
Angela is walking at a constant speed.
You can read her speed in kilometres per hour
from the table: in 60 minutes she walked 4,800 metres
⫽ 4.8 km.
Her speed is therefore 4.8 km per hour.
122
Chapter 4
4
5
6
time in minutes
The graph on the right shows Angela’s walk.
The graph is a straight line passing through 共0, 0兲.
ANGELA'S WALK
metres
4 000
Relationship table
3 000
A relationship table is represented by a
special type of graph, namely a straight-line
graph which passes through (0, 0).
2 000
1 000
0
10
20
30
40
50
minutes
figure 4.23
23 The graph on the right shows Björn’s walk.
BJÖRN'S WALK
a Why does this graph represent a relationship
table?
b Work out the relationship table for this graph,
ending at 60 on the top row.
c What is Björn’s speed in kms per hour?
metres
600
400
200
0
2
4
6
8
10
minutes
figure 4.24
24 Jeroen has been on a bicycle trip.
JEROEN'S BICYCLE TRIP
The graph on the right shows his progress.
What was Jeroen’s speed in kms per hour?
kms
15
10
5
0
5
10
15
20
25
minutes
figure 4.25
A 25
a During a race, Fleur runs 100 metres in 15 seconds.
time in seconds
15
distance in metres
100
60
360
3600
Complete the table and work out her speed in kms per hour.
b Calculate her speed in km per hour if she runs 400 metres in
50 seconds.
Diagrams
123
4.5 Periodic graphs
O 26
At a fairground there is a giant Ferris wheel, a super-swing and a
big dipper. The graphs below show the heights reached by a
fairground visitor on each of these attractions. Which graph
represents which attraction?
height
height
height
time
I
time
time
II
III
figure 4.26
27 The graph on the right shows the periods of
high and low tide at the beginning of June at
Scheveningen.
a At which times did high tide and low tide
occur on 1 June?
b Calculate the period of time between two
successive high tides.
c At which times did the first high tide occur
on 3 June, 4 June and 5 June?
HIGH AND LOW TIDE AT SCHEVENINGEN
high tide
low tide
0 4 9 16.30
5 10 17.30
6 11 18.30
21.30
22.30
23.30
1 June
2 June
3 June
figure 4.27
Theory A
The graph in figure 4.27 repeats itself consistently.
This is because the periods between high and low tide are constant.
Figure 4.28 is another example of a repetitive graph.
This graph repeats itself every 5 minutes.
The 5-minute interval is called the period of the
graph. Graphs like this are called periodic graphs.
height
50
5
40
5
30
20
10
5
0
2
4
6
8
10 12 14 16 18 20 22 24
time in minutes
figure 4.28 This graph shows almost five periods.
Periodic graph
A graph that constantly repeats itself is a periodic graph.
124
Chapter 4
28 Figure 4.29 shows four graphs.
Which of them are periodic? How long are their periods?
6
A
5
B
4
C
3
2
1
D
O
1
2
3
4
5
6
7
8
9
10
time in hours
figure 4.29
29 Figure 4.30 shows part of a periodic graph, whose period is two
.
hours. Add two more periods to it in your
height
5
height
5
4
4
3
3
2
2
1
1
O
1
figure 4.30
2
3
4
time in hours
O
2
4
6
8
time in hours
figure 4.31
30 Figure 4.31 shows one period of a periodic graph.
Add the following three periods to it in your
.
Diagrams
125
31 Paula has been to the fairground. The graph below shows the height
Paula reached when she took a ride on the giant Ferris wheel.
PAULA'S HEIGHT ON THE FERRIS WHEEL
height in metres
20
16
12
8
4
1
2
3
4
5
6
7
8
time in minutes
figure 4.32
a How long is the period?
b At the moment that Paula steps into the red gondola, Anouk is
at the top of the wheel.
.
Draw Anouk’s graph in your
c Stefan is sitting in the blue gondola. Draw a graph showing
as well.
Stefan’s height on the wheel in your
d In the figure below, you can see a billboard fixed to the Ferris
wheel. The p of epn is exactly halfway between the axle and
the highest point of the wheel. Also draw a graph representing
.
the movement of the letter p in your
figure 4.33 Paula steps into the red gondola.
126
Chapter 4
4.6 Diagrams
5
Grijpskerk
Zwaagwesteinde
11
7
3
4
Veenwouden
9
6
7
10
Mantgum
12
or
n
dh
w
ar
de
eu
7
7
n
1
Le
Hurdegaryp
3 Leeuwarden
Camminghaburen
13
Grou-Jirnsum
11
12
Heerenveen of IJsstadion*
m
du
Hoogeveen
K
W rom
or m
og me en
Z rve ieBl aan er Ass
oe d
en
m ijk
de
w
lft
ijk
Gramsbergen
Hardenberg
Enkhuizen
Nieuw Amsterdam
6
8
Kampen
13
12
Dalfsen Ommen
11
Zwolle
n
Wezep
Bu
ite
en
k tru
m
Lelystad Centrum
C
al
7
4
7
28
Hoorn of -Kersenboogerd
Purmerend of
-Overwhere
Emmen Bargeres
5
Dalen
Coevorden
20
Meppel
19
11
pp
St
e
4
Steenwijk
14
11
Emmen
14
Schagen
6
12
Nieuweschans
5 Winschoten
8
Wolvega
9
Heerhugowaard
7
Alkmaar of -Noord
5
Heiloo
7
Castricum
4
Uitgeest
5
Heemskerk 3
7
Beilen
11
13
14
2
Assen
Anna Paulowna
BovenkarspelGrootebroek
of -Flora
Hoogkarspel
4 4
Obdam
10
1
16
4
Den Helder of -Zuid
12
22
5
Workum
5 Hindeloopen
4
Koudum-Molkwerum
Stavoren
4
Groningen
6
Haren
Akkrum
Sneek of -Noord
3
IJlst
13
3
7
4
8
Bedum
4
Sauwerd
Groningen Noord
4
12
Zu
i
H
ar
lin
Fr ge
an n
ek
e
D
ro r
nr
i
D jp
ei
nu
m
4
Harlingen Haven
7
3
3
Winsum
Buitenpost
5
3
5
Baflo
Lo
Usquert
Warffum
U
ith
u
U ize
ith n
u
R ize
oo rm
er
su
de e
m
sc ed
Ap
ho en
pi
ol
ng
e
d
D
am
el
fz
ijl
of
-W
es
t
The map below shows that the distance by train from Meppel to
Assen is 20 ⫹ 14 ⫹ 16 ⫽ 50 kms.
Kr
op
M sw
ar o
t ld
H ens e
of oo ho
Sa ge ek
pp zan
Zu em d-S
ee a
id
b
r pp
Sc roe Oo em
st e
he k
er
em
da
O 32
9
12
11 4
2
2
Heino
14
Mariënberg
Geerdijk
Vroomshoop
Daarlerveen
5 Vriezenveen
figure 4.34 Part of the railway network map.
a What is the distance from Leeuwarden to Grou-Jirnsum?
b What is the distance from Leeuwarden to Groningen?
c How many mms is it on the map from Harlingen to Franeker?
And how many mms are there between Harlingen Haven and Harlingen?
What is the distance in kms between Harlingen and Franeker,
and between Harlingen Haven and Harlingen?
Compare the distances in mms to the distances in kms of these
routes. Does anything strike you?
d Find two other routes whose distances in mms do not conform
with the distances in kms.
e Besides the scale, there are some other differences between
figure 4.34 and a normal map. Name two other differences.
Theory A
The map in figure 4.34 is not a normal map. For example, it only
contains railway lines and the places that have a station.
A schematic map like this is called a diagram.
A diagram is about places and the links between them.
A diagram does not have to be drawn to scale.
Diagrams
127
33 Patricia is spending her holiday on Lanzarote, one of the Canary
Islands. She is staying in Playa Blanca and wants to visit the six
other towns on the diagram. The numbers noted along the links
represent the distances.
Mirador del Rio
Mirador del Rio
Atlantic Ocean
Jameos del Agua
Jameos del Agua
8
25
Costa Blanca
Costa Blanca
Fire Mountains
El Golfo
A
NZ
LA
R
O
T
40
20
E
30
Fire Mountains
6
Arrecife
15
El Golfo
Arrecife
22
15
25
Playa Blanca
Playa Blanca
figure 4.35
figure 4.36
a Name an advantage of the diagram in figure 4.36, compared
with the normal map in figure 4.35.
b Check whether the distance from Playa Blanca to Jameos del
Agua is 55 km.
c You can get to Costa Blanca from Playa Blanca via several
routes.
Name two routes, and work out which is the shorter one.
d How many km is it from El Golfo to Jameos del Agua?
an
Bl
a
ay
128
Chapter 4
ai
ns
G
nt
ou
ca
M
an
re
ta
figure 4.37
ua
Ag
de
l
8
15
38 30
m
48 33
Ar
re
c
ife
Ja
55
eo
s
M
40 25
ira
do
rd
os
C
26 20
el
R
io
Bl
6
Fi
21
35 a Patricia wants to make a round trip starting at Playa
Blanca and stopping at each of the towns once.
Work out the shortest route.
b Patricia goes on the trip.
Would you advise her to take the map, the diagram or
the table with her?
El
15
ol
fo
Pl
figure 4.36. The table shows that the distance between
Playa Blanca and Jameos del Agua is 55 kms.
a What is the distance in kms between El Golfo and
Mirador del Rio?
.
b Fill in the rest of this table in your
ca
34 The distance table on the right belongs with the diagram in
36 Mrs Alink wants to know how to travel by train from
Ommen
0
Zwolle
Winterswijk to Deventer. She would rather use a
diagram than the map in figure 4.38.
a Why do you think she would rather use a diagram
than a map?
10 km
Almelo
Hengelo
Deventer
Apeldoorn
In figures 4.39 and 4.40, you can see two diagrams
representing figure 4.38. Read the information under the
diagrams carefully.
b Which route does Mrs Alink choose, and what is the
distance in kms?
Tw
t
en
e-
ka
na
al
Enschede
Zutphen
IJ
ss
el
Winterswijk
D
Doetinchem
Arnhem
I T
D U
S
L
A
N
figure 4.38 Section of the eastern part of the
Netherlands.
Hengelo
53
Deventer
Deventer
45
16
0–
Zutphen
45
29
1
Winterswijk
63
figure 4.39 Railway lines in the eastern
part of the Netherlands. The numbers
represent the distances in km.
4
Arnhem
Tariff distance
in kms
from – to
Price in Euros
one-way journey
1
3
Zutphen
Arnhem
Hengelo
3
Winterswijk
1
figure 4.40 The numbers represent the
train frequencies.
c On a different occasion, Mrs Alink wants to travel by
train from Arnhem to Hengelo. Dutch Railways advise
her to travel via Deventer. Why do you think they gave
her that advice? How many kms longer is this route
than the shortest one?
d Draw a diagram with the same towns and the same six
links. Mark the price of the train ticket along each
connection.
8
1.40
9– 12
1.90
13– 16
2.50
17– 20
2.90
21– 24
3.40
25– 28
3.90
29– 32
4.40
33– 36
4.90
37– 40
5.30
41– 48
6.10
49– 56
7.00
57– 64
8.00
65– 72
8.90
73– 80
9.90
81– 88
10.80
89– 96
11.80
97–104
12.70
105–112
13.70
113–120
14.40
121–136
15.60
figure 4.41 Prices of one-way train
tickets.
Diagrams
129
37 Of course the points on a diagram don’t always represent towns
Bas
Arie
or villages.
Each point in figure 4.42 represents one of the boys from
first-year class B1E. The link from Arie to Frits means that they
are friends. Where there is no link, the boys are not friends.
Frits
a Is Bas friends with Frits? Is Bas friends with Ed?
Ed
Dick
b Is Frits a friend of Dick’s?
c Who has the most friends, and who has the fewest?
figure 4.42 Diagram of friends.
d How many points and links are marked on the diagram?
e Three friends go swimming one afternoon. Which three
friends could they be?
f Can you find four friends who would want to play together?
If so, who are they?
If not, which extra link would need to be added to the diagram
to make it possible?
Cliff
Theory B
Some of the points in a diagram are connected.
The diagram on the right has six points and eight links.
As you can see, there is no link between A and D.
A link is a connection without steps in-between.
You can choose whether you draw a link as a straight line or a curve.
Be careful, the intersection between AB and EF is not a point on
the diagram.
F
E
Diagrams
C
D
A diagram contains points with links.
Draw a fat dot for each point on the diagram.
figure 4.43
Sometimes there are numbers written along the links.
These can represent distances, times, or something else.
The distance table below is about figure 4.44.
You are travelling from A to C via B, therefore the distance from
A to C is 8 ⫹ 4 ⫽ 12 km.
A
A
B
8
C
12
D
13
B
C
D
8
12
13
4
5
4
5
B
8
A
4
5
C
3
3
3
D
figure 4.44 Distances in km.
130
Chapter 4
B
A
38 The diagram in figure 4.45 is about six boys
from first-year class B1E.
A link between two boys means that they play
the same sport.
a The diagram has six points. One of the points
is not linked to any of the others. Which point
is that? Do you think this boy doesn’t play
any kind of sport?
b The links Arie-Dick and Cliff-Frits intersect,
however the intersection is not a point on the
diagram.
Draw the diagram another way, so that none
of the links intersect.
c The only sport Cliff plays is football.
Do Frits and Ed play football? How do you
know that?
d Dick only plays tennis. Who else plays
tennis?
e Is there another sport played by only two of
the boys? How can you tell?
Bas
Arie
Cliff
Frits
Ed
Dick
figure 4.45 Sports diagram.
39 In the table, you can see which sports five girls from class 1E
participate in.
gymnastics
Louise
×
Marieke
Linda
Bertina
Sylvia
football
tennis
badminton
hockey
×
×
horse riding
×
×
×
×
×
×
×
×
a Draw a diagram for these girls, showing which girls play the
same sports.
b Which do you think is clearer, the table or the diagram?
Diagrams
131
40 When drawing a diagram, you are free to choose where you
place the points. This is why diagrams sometimes look very
different, although they actually mean the same.
a Figure 4.46 contains four diagrams. Three of them mean the
same. Which are they?
b Which diagram is the one that is really different?
a
b
c
d
figure 4.46 Three of the four diagram are not really different.
Theory C
Figure 4.47 shows part of the map of Zutphen. The arrow from A
to B along the Molengracht indicates that this is a one-way street.
On Nieuwstad, you can only go from D to C.
D
250
E
200
200
A
100
250
50
B
C
distances in metres
figure 4.47 Map of part of Zutphen.
The diagram in figure 4.48 applies to the map of
Zutphen. Some of the links in the diagram are marked
with arrows. The numbers along the streets represent
the distances in metres. Where there are no arrows,
you can travel along the links in both directions.
A diagram with one or more arrows is called a
directional diagram.
figure 4.48 A directional diagram of the
map of Zutphen.
to
distance
B
from
C
a Jef Martens wants to go from C to A by
car. How many metres will he have to drive?
figure 4.49
How many metres will he have to drive to get
back to C from A?
b Why is the shortest distance from B to D 350 metres?
c Copy the distance table on the right and fill in the blanks.
Chapter 4
E
B
C
D
250
A
D
132
A
300
350
E
42 a Draw a directional diagram of figure 4.50.
b Calculate the distance from H to G, and
from G to H.
c Calculate the distance from H to B, and
from B to H.
d Between which two points is the difference
between the way there and back the longest?
What is the difference in metres?
figure 4.50 All distances are in metres.
43 At a volleyball tournament, the first-year classes 1A, 1B, 1C, 1D
and 1E are in the same pool.
Halfway through the day, the sports teacher draws the diagram
shown in figure 4.51.
The arrow from 1A to 1B means that 1A has won from 1B.
a There is no arrow marked yet between 1B and 1D. What does
this mean?
b How many matches has 1B played? How many matches has
1B won?
c Which class has played the fewest matches?
1A
1B
1C
1E
1D
figure 4.51 Diagram of the volleyball tournament.
figure 4.52 Table of results.
d Fill in the table in figure 4.52 in your
.
e At the end of the sports day, each class has
played once against every other class. How
many matches were played altogether?
f Could class 1A win the tournament? Explain
your answer.
Diagrams
133
A 44
to
The distance table on the right requires a
directional diagram consisting of six links, two of
which are one-way streets.
, including
Draw this diagram in your
the distances for each link.
distance
A
A
from
B
C
D
E
70
70
50
30
10
30
40
20
40
B
30
C
70
80
D
50
60
20
E
30
40
20
20
20
figure 4.53
The seven bridges of Königsberg
HISTORY
18th century Königsberg was divided into four parts
which were linked by seven bridges, as shown on the
map. The inhabitants asked themselves whether it was
possible to make a round trip by crossing each bridge
only once. Whichever way they tried it, they failed.
C
C
KNEIPHOFF
D
A
B
D
A
B
a
b
figure 4.54 The four parts of the city with their seven
bridges.
The mathematician Euler demonstrated that such a trip was, indeed, impossible.
He used the diagram shown in figure 4.54b, in which the links represent the seven
bridges. If you want to know more about this conundrum, surf the Internet using the
key word Königsberg.
134
Chapter 4
4.7
Diagrams using the Dutch Railways (NS)
route planner
45 Temporary employment agency GAMS has branches in Amsterdam, Middelburg,
Groningen and Sittard. Employees regularly travel between the branches.
The conditions are that the first train leaves after 9 a.m. on Mondays, and that
the train tickets are second-class, reduced rate.
a Start up the NS route planner, and then use it
to answer the following questions.
b Onno leaves Middelburg to travel to
Groningen.
1 What is the departure time of the train?
2 How much does a one-way ticket from
Middelburg to Groningen cost?
3 What is the distance in kms from
Middelburg to Groningen?
4 How often does Onno have to change trains?
5 How much more would it cost Onno if he
bought a one-way ticket MiddelburgRotterdam and a one-way ticket Rotterdam-Groningen?
c Draw a diagram with the points Amsterdam, Middelburg, Groningen
and Sittard, and draw in the six links. Note the cost of a one-way
ticket for each link. You have drawn a ‘price diagram’.
d Draw a ‘travelling-time diagram’ for the four cities.
e Draw a ‘frequency-of-changes diagram’.
GAMS’ biggest competitor is ZMAK, which has branches in Zwolle, Kampen, Lelystad,
Amersfoort, Utrecht, Arnhem and Deventer.
Jasmin wants to make a diagram of these branches. She uses the NS route planner to
see whether she can travel between any two points without having to change trains.
Where it is possible, she draws a link between the relevant two points, under the
same condition that the first train leaves after 9 a.m. on Mondays.
.
f Complete the diagram in your
You have now drawn a ‘no-changes diagram’.
g One of the points in the ‘no-changes diagram’
is not linked to any of the other points. Which
point is this? What does this imply?
h Martijn looks at Jasmin’s diagram and
believes that he can travel from Utrecht to
Zwolle without having to change at
Amersfoort.
1 Why does Martijn believe this?
2 Check with the NS route planner whether
he is right.
i Draw a ‘one-change diagram’ in your
.
Diagrams
135
4.8 Summary
§4.1
p. 110
Pictograms and bar charts
Tables, bar charts and pictograms are used to represent all
kinds of information in an orderly way.
To make a bar chart you should
• place a heading at the top
• draw the bars separately
• name the subject of each axis.
TEMPERATURE IN A CLASSROOM
Rises and falls
The graph on the right is a general graph.
The axes of a general graph carry no
numbers. You can’t read precise
temperatures on it, because it is only meant
to show the course of the graph.
The section of the graph pointing upwards
is said to be rising. The section pointing
downwards is said to be falling. The
horizontal section is said to be constant.
§4.3
p. 117
ng
p. 113
constant
risi
§4.2
temperature
fa
llin
g
ng
isi
r
time
figure 4.55
Drawing graphs
You can draw a graph to represent a table.
• Draw the horizontal axis using the data from the top row
of the table.
• Draw the vertical axis using the data from the bottom row
of the table.
• Name the subject of each axis.
• Plot the points from the table.
• Draw the graph by joining all the points.
A graph is often shown as a flowing curve. Some graphs consist of straight
horizontal lines or floating points. This depends entirely on the situation.
JORIS' WEIGHT
MUSEUM ENTRANCE FEE
PRICE OF FRUIT TREES
kg
30
Euros
15
Euros
30
25
20
20
10
15
10
10
5
5
0
figure 4.56
136
Chapter 4
3
6
9
age
2000
2001
2002
year
0
1
2
3
4
5 6
quantity
§4.4
Graphs and relationships
p. 122
The graph of a relationship table is a straight line through
共0, 0兲. This type of graph is typical of, for instance, the
rotation of a ship’s screw, distances covered at a constant
speed, or exchanging money.
§4.5
Periodic graphs
Graphs that repeat themselves are called periodic graphs.
The time during which one cycle occurs is called a period.
p. 124
§4.6
p. 127
Diagrams
Diagrams and tables
Figure 4.57 shows a diagram of five localities close to Arnhem.
Figure 4.58 is the distance table represented by the diagram.
Posbank
6
2
g
6
ee
4
St
k
an
rg
3
be
5
en
6
Velp
4
Rheden
2
De Steeg
figure 4.57 Diagram of the Veluwezoom.
4
3
Zi
jp
6,5 2,5
Po
sb
2,5
e
3
D
4
R
he
de
n
Ve
3
lp
Zijpenberg
figure 4.58 Distance table of the Veluwezoom.
The distance Velp-Posbank is 4 ⫹ 2.5 ⫽ 6.5 km.
You are looking for the shortest route, which is via Rheden.
The diagram in figure 4.57 contains five points and seven links.
p. 130
Diagrams
• A diagram consists of points and links.
• Use fat dots to draw the points on the diagram.
• The points can represent all sorts of things, such as cities,
classes, or people.
• A diagram does not have to be drawn to scale.
p. 132
Directional diagrams
A directional diagram contains one or more
arrows.
In figure 4.59, the arrow between D and A
means that you can go from D to A, but not
from A to D.
You can go in both directions between B
and D.
The distance from C to D is 3 km, and the
distance from D to C is 8 km. 共To go from
D to C, you have to go via A.兲
A
B
3
7
5
D
3
C
4
figure 4.59 Distances in km.
Diagrams
137
Diagnostic test
.
test on the
§4.1
1
At 8 p.m., at the end of each working day,
STOCK OF SOFT DRINKS
bottles
the manager of KOKO supermarket asks his
1 000
1 staff to count the stock of soft drinks. The
results are shown on the bar chart on the
800
right.
a How many bottles of soft drink were
600
counted on Tuesday evening?
b On which evening were 450 bottles
400
counted?
c How many bottles were sold on Tuesday?
d Once during the week, the stock of bottles 200
was replenished to 1,000.
On which day was this? How many
Mon
Tues
Wed
bottles were sold that day?
figure 4.60
e On which day were the most bottles sold?
How many?
revision exercise
§4.2
Thurs
Fri
2
Below, there are four sentences and four general graphs.
a Match the correct graph with the correct sentence.
2
A At the end of his bicycle trip, Joop had a following
wind.
B Mirella had a flat tyre on her way home.
C Harmen cycled at a constant speed.
D Along the twisting dyke road, Elsa alternately had a
following or a head wind.
revision exercise
speed
speed
speed
time
I
speed
time
II
time
III
figure 4.61
, colour in all the sections of the
b In your
graphs that rise.
c What is graph I called?
138
Chapter 4
time
IV
Sat
D
3
The greengrocer knows all this summer’s prices of
strawberries and grapes. His daughter Petra has
3 drawn the graphs shown in figure 4.62 to represent
those prices.
a In which month were the strawberries the
cheapest?
b What was the price per kilogram then?
c In which month was the price of strawberries and
grapes the same?
d During which period were the strawberries
cheaper than the grapes?
revision exercise
PRICES PER KG
Euros
5
4
strawberries
3
2
grapes
1
May
June
July
Aug.
Sept.
figure 4.62
§4.3
4
During training, Tom measures his heart rate. First, he
sprints for one minute, and then walks until his heartbeat
4 has dropped to 100. The measurements are listed in the
table below.
a Draw a graph to represent the table. Decide how to suitably number the axes yourself.
b His heartbeat rises more quickly than it falls. How can
you tell that from your graph?
c How many times did Tom sprint during those eight
minutes?
d Does his heart rate fall faster or more slowly after his
second sprint than it did after the first?
HEART RATE
revision exercise
Heart rate has to do with
the speed at which the
heart pumps blood around
the body. An adult’s heart
rate varies between 60 and
80 beats a minute. Hard
physical effort will make it
rise quickly.
TOM’S TRAINING
time in minutes
0
1
2
3
4
5
6
7
8
heart rate
70
145
120
100
155
130
105
100
160
Diagrams
139
5
THORBECKE POST in Zwolle charges the
postal tariffs shown on the right.
5 a How much does it cost to send a letter
weighing 10 grams? And one weighing
40 grams?
b Wilfred wants to send two letters. He
has to pay 160 cents.
How heavy could his letters have been?
revision exercise
POSTAL RATE AT THORBECKE POST
tariff in cents
120
100
80
60
CAPELLEN POST
advertises the following
postal tariffs.
Letters
0– 40 grams
40 cents
40– 80 grams
70 cents
80–120 grams
120 cents
c Draw a graph representing the
CAPELLEN POST postal tariff.
d Wilma has two letters, one of 30 grams,
the other of 68 grams. She wants to go
to only one post office to send them.
Which post office will she choose?
§4.4
6
For 1 Euro you can get 90 Icelandic
Crowns.
6 a Copy the relationship table and fill in
the blanks.
b Draw a graph to represent the table.
c The entrance ticket to the national
museum in Reykjavik costs
200 Icelandic Crowns.
How many Euros is this?
revision exercise
§4.5
7
The graph in figure 4.64 shows the water
levels at Vlissingen.
7 a This graph is a periodic graph.
How many hours is one period?
b Add a further period to the graph in your
.
c High tide occurs three hours later at
Zandvoort.
Also draw the water levels graph for
Zandvoort, between 0 and 24 hours.
40
20
O
20
40
60
80
100
120
weight in grams
figure 4.63
EXCHANGE TABLE
EUROS – ICELANDIC CROWNS
euros
1
10
25
60
5,850
crowns
11,250
HIGH AND LOW TIDES
revision exercise
high
tide
low
tide
0
figure 4.64
140
Chapter 4
3
6
9
12
15
hours
D
§4.6
8
Figure 4.65 shows a map of De Hoge
Veluwe National Park.
8 The major paved roads are shown in
yellow.
The distances are in kilometres.
a Draw a diagram with five points to
represent the map. Add the distances.
b Work out a distance table from the
diagram, as shown in figure 4.66.
c A car trip starts at Rijzenburg, and returns
there via the other four locations.
What is the distance of the shortest
possible route?
revision exercise
St. Hubertus 3
Hunting Lodge
4
Otterlo
2.5
Museum
Kröller-Müller
GE VELUWE
DE HO
8
Rijzenburg
Entrance
rlo
lo
tte
oe
nd
er
O
Hoenderlo
3.5
3
R
ijz
en
bu
Kr
rg
öl
le
rM
.
H
3
ub
er
tu
s
H
figure 4.65 De Hoge Veluwe
figure 4.66
9
Look at the map in figure 4.67.
a Draw a directional diagram with 5 points.
9 b Calculate the distance from A to B, and also from B to A.
c Work out a distance table for this diagram, as shown in
figure 4.68.
revision exercise
to
distance
A
B
C
D
E
A
B
from
C
D
E
figure 4.67 Map. Distances in metres.
figure 4.68 Distance table.
Diagrams
141
Revision
§4.1
§4.2
1
2
All the pupils of a first-year class have been
talking about where they are going to spend
their holidays. Look at the bar chart on the right.
a How many pupils are going to France for their
holidays, and how many to Spain?
b How many first-year pupils are not going
away for the holidays?
c How many children are going abroad?
d How many first-year pupils are there
altogether?
e Theresa is going to Italy. Which bar does she
belong to?
f Which countries are being visited by the same
number of pupils?
HOLIDAY DESTINATIONS
6
5
4
3
2
1
0
Nether- France Spain
lands
Ger- Turkey Other Not going
many
Countries away
figure 4.69
Figure 4.70 below represents a conversation between three people.
There are three general graphs. Match the right person to the right graph.
A Dick: ‘With my diet, I first lost weight, but now I’m putting it on again fast.’
B Anne: ‘My baby is growing like a weed.’
C Joop: ‘I lost a lot of weight while I was ill.’
weight
weight
weight
time
I
time
time
II
III
figure 4.70
3
Look at figure 4.71.
a How many degrees was it in Amsterdam and
in Brussels at 4 a.m.? By how many degrees
did the temperature differ?
b What was the difference in temperature at
6 a.m.?
c At what times was the temperature the same
in Amsterdam as in Brussels?
d For how long was the temperature in Brussels
below zero on that day?
TEMPERATURE CURVES
degrees C
10
Brussels
5
O
–5
–10
figure 4.71
142
Chapter 4
Amsterdam
4
8
12
16
20
time
24
R
§4.3
4
The pupils of B1G are making a bean report for biology.
Pedro places a bean on a damp sponge. The bean sprouts and a
plant grows from it. Pedro makes a table of the height of his
bean plant. You can see the result below.
HEIGHT OF PEDRO’S BEAN PLANT
time in weeks
0
1
2
3
4
5
6
7
8
height in cms
0
2
3
5
9
13
18
20
21
a Draw a graph to represent this table. Place the time along the
horizontal, and the height along the vertical axis.
b In which week did the plant grow the fastest? How can you
tell that from your graph?
5
Figure 4.72 below shows the price of a submarine roll at the
PIRAMIDE snack bar during the course of 2002.
PRICE OF SUBMARINE ROLLS IN 2002
1.80
1.70
1.60
1.50
j
f
m
a
m
j
j
a
s
o
n
d
time
figure 4.72
a How much did a submarine roll cost in September?
And how much in November?
b During which months was a submarine roll the cheapest?
And when the most expensive?
c Which was the longest period during which the price remained
the same?
d At snack bar KEGEL, a submarine roll always costs
15 eurocents less than at PIRAMIDE.
Draw a graph to represent KEGEL.
Diagrams
143
§4.4
6
At STOPHIER petrol station, petrol costs € 1.10 per litre.
The relationship table below applies.
no. of litres
price in Euros
1
5
1.10
5.50
9
15
30
23.65
a Copy the table and fill in the blanks.
b Draw a graph to represent the table.
Select suitable numbering for the axes.
c Rob de Bruin had to pay € 35.75 at the petrol station.
Read your graph to ascertain how much petrol he tanked.
Check your answer by calculating it.
§4.5
7
a Which of the graphs below are periodic?
b Add another period to the periodic graphs in your
2
2
1
1
O
1
2
3
4
5
O
6
1
.
2
graph A
2
2
1
1
O
1
2
3
3
4
5
6
4
5
6
graph B
4
5
O
6
1
2
graph C
3
graph D
figure 4.73
§4.6
8
Figure 4.74 shows a map of WOESTE GRONDEN Nature Reserve.
The cycle paths are indicated in red.
The numbers represent the distances in kilometres.
The diagram in figure 4.75 represents the map.
D
D
8
8
E
E
5
4
A
F
2
A
7
5
3
B
C
C
figure 4.74 Map of Woeste Gronden showing cycle paths.
144
Chapter 4
F
B
figure 4.75 Diagram of Woeste Gronden’s cycle paths.
R
a Copy the diagram in your exercise book and add the distances
A
of the links in km.
B
b The distance from D to F is 5 kms. You can see this
number in the blue square of the distance table.
C
Copy the distance table and write the distance from C to E
D
in the green square.
c Work out the distance from A to D as follows. Look for the
E
shortest route from A to D, which is via F, since 2 ⫹ 5
5
is less than 8.
Therefore, write 7 in the red square.
d Which is the shortest route to get from A to E?
Write the distance in the yellow square.
e Fill in the rest of the table.
f What is the shortest distance of a bicycle trip that begins at A,
passes all the other points, and ends at C?
9
F
The map in figure 4.77 shows a number of one-way streets.
You can, for instance, go from A to B, but not directly from
B to A.
The shortest route from B to A is via C and F. The route
is 120 ⫹ 40 ⫹ 100 ⫽ 260 metres long.
to
distance
A
100
F
80
E
40
40
from
B
C
D
E
F
60
A
B
60
A
260
C
D
B
120
C
70
D
E
F
figure 4.77 Map. Distances are in metres.
a How long is the route from E to F? And from F to E?
b Draw a directional diagram to represent this map. Why do you
have to draw arrows on some of the links?
c The distance table in figure 4.78 applies to the diagram.
Some of the numbers have already been filled in.
Explain these numbers.
d Copy the table and fill in the blanks.
Diagrams
145
Extra material
Solar time
1
Many people believe that the sun stands exactly in the south at 12 midday. This is not correct.
Since the Second World War, the Netherlands has applied Central European Time 共C.E.T.兲.
The graph shows at which times the sun is in the south above Utrecht according to C.E.T.
As you can see, the time varies considerably during the course of a year.
AT WHICH TIMES IS THE SUN IN THE SOUTH ABOVE UTRECHT?
time (winter time)
13h 00m
time (D.S.T.)
14h 00m
50m
50m
40m
40m
30m
30m
12h 20m
Jan.
Feb.
Mar.
Apr.
May
June
July
Aug.
Sept.
Oct.
Nov.
Dec.
13h 20m
figure 4.79 Daylight saving time is from the last Sunday in March until the last Sunday in October.
a At which time is the sun in the south on
1 January?
b At which time is the sun in the south on
1 September?
c Joyce says that the sun is in the south later on
1 August than on 1 February.
Explain why her observation is correct.
d On approximately which date is the sun in the
south at 12.35 p.m.?
e On approximately which dates is the sun in
the south at 13.42 p.m.?
By 1900 the Netherlands still had its own time,
which was 40 minutes behind C.E.T. winter
time. Daylight saving time didn’t exist yet.
f On approximately which dates in 1900 was
the sun in the south at exactly 12.00 midday?
146
Chapter 4
All clocks to be set forward
by 1 hour and 40 minutes
The A.N.P. made an official announcement
yesterday:
By order of the Supreme Command of the
German occupying forces, it is announced
that until further notice and with immediate
effect, the population must apply the same
black-out regulations throughout the
Netherlands as have been in force during the
last few days.
In order to minimise the effects of this order
as much as possible during the coming days,
it is further ordered that at 12 midnight
tonight, the same time 共daylight saving time兲
will be introduced in the Netherlands as in
Germany. Therefore all clocks must be set
forward to 1.40 a.m. at midnight.
The above decision also has advantageous
economic consequences, as almost two extra
hours of daylight are gained in the evenings.
De Telegraaf newspaper, 16 May 1940
E
Snow depths
2
During a severe frost, a thick layer of snow acts like a blanket.
The graph below shows the relationship between the temperature
of the air, the depth of the snow, and the ground temperature.
You can see, for example, that with a layer of snow 20 cm deep
and an air temperature of ⫺15°, the ground temperature is ⫺8°,
as shown by the green dot.
depth of snow in cm
air temperature
60
-10
50
-15
40
-20
30
-25
20
-30
10
0
-2
-4
-6
-8
-10
-12
-14
-16 -18 -20
ground temperature
figure 4.80
a What is the ground temperature if the air temperature is ⫺25°,
and the layer of snow is 30 cms deep?
b If the air temperature is ⫺30° and the ground temperature is
⫺10°, how deep is the snow?
c In areas such as Siberia, farmers often shovel extra snow onto
the fields. How can you tell from looking at the graph that this
is a sensible thing to do?
d The air temperature is ⫺30°. The snow is 15 cms deep.
Farmer Wassily adds an extra 40 cms thick layer of snow. By
how many degrees will the ground temperature rise?
e At a given time, the air temperature is ⫺20° and the ground
temperature is ⫺15°.
Is this the right time for Boris to go on a langlauf?
Diagrams
147
Cable cars
3
Cable cars are often found in the Alps. They transport you from a
station in the valley to a much higher mountain station, and back
again. Most of them have two cabins, I and II. The cabins are
suspended from strong steel cables. The two cabins leave the
valley and mountain stations respectively at the same time.
The graph below applies to car I.
CABLE CARS IN THE ALPS
height
mountain
station
I
valley
station
9
10
11
time in hours
figure 4.81
a How many minutes does cabin I take to travel
from the valley to the mountain station?
b At the mountain station, people get off and
others get on. How much time do they have?
c The graph of car I is periodic. How many
minutes is a period?
, draw the graph for
d In your
cabin II.
e The cable cars operate from 9 a.m. until
12 midday in the mornings, and from 1 p.m.
until 4 p.m. in the afternoons.
How many times a day do the two cabins pass
each other?
At which times?
148
Chapter 4
E
The diagram game
4
The rules of the diagram game are shown below.
Rules of the diagram game
1 Draw three points.
2 Draw a link. This can be done in one of two ways.
Link two points with a curve.
Draw a loop joining the point to
itself.
Then add another point to the curve or loop you have just drawn.
3 In turn, your opponent has to draw a curve or a loop, and also add
an extra point.
When you continue playing, think of the following rules.
4 A link is not allowed to intersect another one.
WRONG! Two links are
intersecting.
WRONG! More than three links
to one point.
5 Not more than three links may converge in one point.
6 The one who succeeds in adding the last link, wins the game.
a Play this game with a fellow pupil a few times.
Does the first player always win?
b You can also play the game by starting with four or more
points. Why don’t you try it.
Diagrams
149
chapter
05
Lines and angles
You must be very careful when cycling close
to a truck, because the driver is not able to
see certain parts of the road. This is because
of the ‘blind spot’ in his viewing range.
Annually, thirty to fifty traffic fatalities occur
because the driver was unable to see a
cyclist due to the blind spot.
• Why does a reversing truck give a warning
sound signal?
• Do you think there is also a blind spot on
the left of the truck?
• Is the blind spot larger or smaller with a
convex mirror than a flat one? Explain.
The government is currently imposing strict
regulations on truck drivers’ viewing range,
by for instance making an additional mirror or
a camera and monitor compulsory.
• The computer program ‘Van groot gewicht’ (This carries weight)
will help you to find out how to adapt your behaviour as a cyclist,
to ensure your safety in traffic.
150
Chapter 5
directly
visible
blind
spot
visible in
the mirror
what you can learn
ICT
– In this chapter you will learn all about perpendicular
and parallel lines. You will learn how to draw them
with your protractor triangle.
– You will also draw angles and measure them.
– Angles are measured in degrees.
– A right angle has 90 degrees.
– Computer program ‘Van groot gewicht’ (This
carries weight) on page 150.
– Computer program ‘Hoeken’ (Angles) on page 164.
– Learning: digital learning line on page 175 with the
applet Turning.
paragraph.
5.1 Lines
O
1
There are a number of expressions that include
the word ‘line’
a what is the meaning of
– direct line descendant
– line-up
– toe the line
b Think of some other expressions that include
the word ‘line’.
O
2
Both horizontal and vertical lines may occur in
space shapes. In the photograph on the right,
they may not be immediately apparent, but they
really are there.
a Are the red lines horizontal or vertical?
b Are the blue lines horizontal or vertical?
c Do the blue and red lines form a rectangle in
reality?
d The window cleaner is standing on a platform. figure 5.1
Is the platform suspended horizontally?
O
3
In the figure on the right, lines l and m are at right angles to
each other.
a Check this with the right angle of your protractor triangle.
b Are the lines k and m also at right angles?
c How about the lines m and p?
p
k
l
m
figure 5.2
Theory A
7
6
When we talk about lines, we mean straight lines.
A line has no beginning or end point.
A line is indicated with a small letter.
4
20
160
3
2
1
1
30
150
0
2
1
4
3
50
130
1
figure 5.3
0
16 20
Chapter 5
0
17 0
1
152
6
2
15
30 0
130
50
120
60
110
70
100
80
ER
ME
ER
ET
ZO
3
140
40
6
12 0
0
5
7
40
140
2
3
The lines l and m in figure 5.2 are at right angles.
We say: ‘l is perpendicular to m’, or ‘l is
a perpendicular of m’.
You need a protractor triangle on which the printed
perpendicular is marked in red, as shown in figure
5.3, to draw a perpendicular line.
10
170
5
90
80 0
10
70 10
1
How to draw a perpendicular through a point on a line
I
4
100
80
90
I
170
10
0
16
20
0
15
30
3
60 0
12
5
6
2
3
70 0
11
1
110
70
1
3
2
1
170
10
0
16
20
0
15
30
7
6
5
4
14
40 0
3
13
50 0
2
120
60
90
100
80
110
70
120
60
3
13
50 0
14
40 0
0
15
30
100
80
0
70 0
11
MEER
ZOETER
80
100
2
1
0
1
2
2
1
3
4
5
0
16
20
7
6
170
10
I
90
MEER
ZOETER
80
100
2
2
3
1
1
60 0
12
10
170
20
160
30
150
3
40
14
0
4
50
13
0
5
6
4
7
10
170
20
160
30
150
P
40
14
0
Slide your protractor triangle along until
you reach point P, and then draw the
perpendicular. Extend the line somewhat
at both ends, for the perpendicular does
not end at P or l.
50
13
0
2
7
7
6
5
4
14
40 0
3
1
2
13
50 0
3
2
120
60
1
110
70
0
1
MEER
ZOETER
80
100
2
70 0
11
1
3
2
3
P
60 0
12
10
170
20
160
30
150
5
6
7
Align the perpendicular printed on
your protractor triangle with line l.
40
14
0
1
P
50
13
0
Draw the perpendicular m through P
on line l as follows.
3
Draw the
symbol in the right-hand corner.
Write the letter m next to the line.
P
I
m
4
5
Look at figure 5.4.
a In your
, draw line m through
P, perpendicular to line l.
symbol.
Don’t forget the
b Draw line n through Q, perpendicular
to l.
c Draw line p through R, perpendicular
to l.
P
Q
R
l
figure 5.4
Look at figure 5.5.
, draw the
a In your
perpendiculars to all the sides of the
parallelogram through P.
b The perpendiculars divide the parallelogram
into 4 sections. Cut out the sections and
reassemble them in the shape of a rectangle.
P
figure 5.5
Lines and angles
153
O
6
Masons use a plumb line, which is a piece of
string with a weight attached to one end.
a What does a mason use a plumb line for?
b What would you use a spirit level for?
c What do you use a set square for?
7
a Plot the following points on an axis:
A共⫺1, 3兲, B共2, 1兲 and P共⫺1, 0兲.
b Draw line s through points A and B.
Don’t forget to add the letter s.
c Draw line t through P, perpendicular to
line s.
8
Sometimes it looks as if two
lines don’t intersect, however,
if you extend them, you will
find that they do.
a Which of the pairs of lines
intersect?
b Draw the intersection in your
.
figure 5.6
b
d
a
c
e
f
g
h
figure 5.7
O
9
The extract below is from a book by Bill Bryson.
A single track stretched before us, two parallel steel
rails, perfectly straight and glistening in the sun,
linked by an infinity of horizontal concrete sleepers.
Somewhere along the insanely distant horizon, the two
shining steel lines met at a shimmering vanishing
point.
Endlessly and monotonously, we sucked up the
sleepers, but however far we travelled, the vanishing
point remained stationary. You, or at least I, couldn’t
look at it without getting a headache.
154
Chapter 5
‘How far is it to the next bend?’ I asked.
‘Three hundred and sixty kilometres,’ Willis answered.
‘Doesn’t this drive you crazy?’
‘No,’ they answered with one, seemingly honest, voice.
‘Do you ever see anything that breaks the monotony –
like animals, or something?’
‘A few kangaroos,’ said Coad. ‘A camel now and again.
Sometimes, someone on a motorbike.’
a
b
c
d
Which country is the setting for this story?
The text contains the word ‘parallel’. What does it mean?
What does the author mean by ‘the two shining steel lines’?
Do the steel lines really meet?
Theory B
The lines l and m on the right do not intersect, even if you extend
them at both ends. Lines like this are called parallel lines.
l
Parallel lines
m
Parallel lines go in the same direction.
They do not intersect each other.
figure 5.8
How to draw parallel lines
A
Draw line k through point A parallel
to line l as follows:
l
5
4
3
2
1
0
1
1
2
160
20
3
150
30
0
14 0
4
0
13 0
5
Draw line k. Add an arrow on each line.
2
3
3
12
60 0
4
5
1
2
ZOETERM
11
70 0
100
80
90
EER
80
100
70
110
60
120
6
10
170
6
7
2
16 0
0
7
170
10
2
A
Place the long side of your protractor triangle
alongside point A, in such a way that l lines
up with one of the printed parallel lines on your
protractor triangle.
3
15 0
0
1
50 0
13
40 0
14
l
A
k
l
The arrows show that
the lines are parallel.
, draw a line through P
a In your
parallel with line m, and call it n.
b Draw line k through point Q, parallel with
line m.
Q
m
P
figure 5.9
Lines and angles
155
A 11
a Plot the points A共3, 0兲, B共0, 4兲, C共5, 4兲, D共2, 4兲 and
E共⫺1, 1兲. Draw line p through points A and B.
b Draw line k through C, perpendicular to line p.
c Draw line l through D, parallel with line p.
d Draw line q through E, perpendicular to line p.
e Do you think line l is perpendicular to line q? Why?
A 12
a
b
c
d
Draw line l through points O and A共6, 2兲.
Draw line m through B共3, 1兲, perpendicular to line l.
Draw line n through A, parallel with line m.
Do you think that line l is perpendicular to line n? Why?
13 In figure 5.10, you can see three lines.
a How many intersections are there?
k
k
n
l
l
m
m
figure 5.10
figure 5.11
b Another line, n, has been added in figure 5.11. How many
intersections are there now?
, add a further line in such a way that
c In your
you create as many intersections as possible.
You can carry on with this. With every line you add, make sure
you create as many intersections as possible.
d Copy the table below and fill in the blanks.
no. of lines
2
3
4
5
6
no. of intersections
e Can you see a regular pattern in your table?
f Martijn has drawn seven lines with as many intersections as
possible.
Can you tell how many intersections there are without
drawing all the lines?
g Hanneke has drawn a lot of lines on a piece of paper, with as
many intersections as possible. Altogether, she has counted
45 intersections.
How many lines did Hanneke draw?
156
Chapter 5
14 Margriet has found a way of cutting a triangle
into three sections and reassembling them into a
rectangle. She calls this method the parallelperpendicular-cutting method.
a Check whether Margriet’s method can be
.
applied to the triangles in your
b Kees says: ‘I can cut any quadrilateral into six
pieces and reassemble the pieces in the shape
of a rectangle. First, I cut along a diagonal,
and then . . .’.
Complete Kees’ sentence.
c Check whether Kees’ method can be applied
.
to the quadrilaterals in your
d Margriet says that Kees’ method doesn’t
always work. Try it with the quadrilateral
.
PQRS in your
What went wrong?
Margriet’s method
C
A
B
1 Cut through the centre of the shortest side,
parallel with the longest side.
C
N
M
A
B
2 From C, make a vertical cut to MN.
C
M
M
N
N
A
B
3 Your rectangle is complete.
Lines and angles
157
5.2 Angles
O 15
There are many expressions in Dutch that contain the Dutch
word for angle, ‘hoek’ 共which, by the way, also means ‘corner’兲.
figure 5.12
O 16
Figure 5.13 is the view of a truck as seen from
above. In his wing mirrors, the driver can only
see the sections of the road that are shaded in
blue. One of the mirrors is flat, while the other
one is convex. On which side of the truck is the
convex mirror mounted? Explain your answer.
figure 5.13 Areas visible in the wing mirrors.
158
Chapter 5
17 Tabitha is paddling along a canal in her canoe. In front of her, in between
the high quay walls, she can see a number of mooring bollards. On the left
you can see the situation at 10.00 a.m. On the right, it is one minute later.
figure 5.14
a How many bollards could Tabitha see at 10.00 a.m.?
.
First, draw the lines of vision in your
b How many bollards could she see a minute later?
c When was the viewing angle wider, at 10.00 a.m. or at 10.01 a.m.?
Theory A
The lines of vision you drew in exercise 17a form an angle. The viewing angle
is wider at 10.01 a.m. than at 10.00 a.m. The angle enclosed by lines of vision is
just one example of an angle. In fact, there are angles all around you.
figure 5.15
When the ladder rises, the angle widens. When the ladder is vertical,
it is standing at right angles.
Lines and angles
159
18 Figure 5.16 is a bird’s-eye view of a castle wall with an opening.
a The observers at A, B, and C are looking through the
, using different colours, draw
opening. In your
the three viewing angles.
b Which viewing angle is the smallest?
figure 5.16
figure 5.17
19 The hands of a clock are at an angle to each other.
a What is the time when they are at right angles?
Write down two possibilities.
b When is the angle of the hands wider: at 2 o’clock or at
5 o’clock?
c Why aren’t the hands at rights angles at a quarter past six?
d The small hand turns around the clock once every 12 hours.
How often does the small hand form a right angle with the
large hand during that period?
20 A knowledge of angles is necessary for a number of occupations.
For which of the following occupations does this apply?
Explain your choices.
chair designer
French teacher
pilot
dentist
160
Chapter 5
cashier at Albert Heijn
greengrocer
mason
architect
Theory B
Every angle has two sides. Both sides begin at the
vertex. It doesn’t matter how long the sides are, the
angle doesn’t change.
A vertex is usually indicated with a capital letter.
e
sid
P
side
vertex
In the figure below, angle A is a right angle.
symbol.
You can see that from the
Angle B is smaller than a right angle.
Angle B is an acute angle.
Angle C is larger than a right angle.
Angle C is a wide angle.
The sides of angle D form a straight line.
The angle is therefore a flat angle.
e
sid
P
side
vertex
figure 5.18 These two angles are the same size.
A
right angle
B
acute angle
C
wide angle
D
flat angle
figure 5.19
21 Figure 5.20 contains seven angles.
a
b
c
d
Which of the angles are acute?
Which of the angles are wide?
Which of the angles are right angles?
Which angle is flat?
22 a Draw a rectangle ABCD with AB ⫽ 5 cm
and BC ⫽ 3 cm.
b How many right angles can you spot?
c Draw the diagonal AC.
How many acute angles can you see?
d Draw the diagonal BD. The intersection of
the diagonals is S. How many wide angles
are there at S?
e How many acute angles are there in your
drawing?
A 23
figure 5.20
Fill in right, acute, wide or flat where applicable.
What type of angle do the hands of a clock form at the following times?
a at 3 o’clock they form a . . . angle
d at a quarter past 12 they form a . . . angle
b at 6 o’clock they form a . . . angle
e at a quarter past nine they form a . . . angle
c at 5 o’clock they form a . . . angle
f at 11.28 they form a . . . angle
Lines and angles
161
5.3 Measuring angles
O 24
Figure 5.21 shows four acute angles. Place them in the correct
sequence, from the most to the least acute angle.
A
B
C
figure 5.21
Theory A
All sorts of things can be measured. By measuring
you can find out: that the traffic jam on the A12 is
13 kms long, that the traffic jam on the A28 will cost
you 28 minutes, that the content of an aquarium is
300 litres.
Angles can also be measured.
They are measured in degrees.
A right angle is 90 degrees wide. A right angle can be
divided into 90 equally large angles of 1 degree each.
Instead of 90 degrees, we write 90°.
figure 5.22 This right angle is divided into 90 degrees.
The red angle in figure 5.22 is 18°. Check it.
162
Chapter 5
D
The angle drawn below is 1° wide, which is really small.
figure 5.23
25 a Take a loose sheet of paper, draw a square on it whose sides
measure 8 cm, and then cut it out.
b Fold it in half along a diagonal. You now have a triangle.
How many degrees are the acute angles of the triangle?
c Your protractor triangle has three corners. How large are the
angles of these corners?
26 a How many degrees are there between the
hands of a clock at 3 o’clock?
b How large is the angle at 1 o’clock?
c How large is the angle at 4 o’clock, and at
6 o’clock?
figure 5.24
27
Which number could Sarah have mentioned?
28 a Fill in 0°, 90° or 180° where applicable.
An acute angle measures between . . . and . . .
A wide angle measures between . . . and . . .
b If you are sure that you have answered question a correctly,
write these two sentences in your exercise book. Colour them
with a marker.
Lines and angles
163
29 a How many degrees does the small hand of a clock travel in an hour?
b
c
d
e
f
g
h
How many degrees does the small hand travel in 10 minutes?
How many degrees does the large hand travel in 10 minutes?
How large is the angle between the hands at ten past twelve?
How large is the angle between the hands at ten to twelve?
How large is the angle between the hands at half past five?
How large is the angle between the hands at six minutes past one?
How large is the angle between the hands at 15.48 hrs?
The program ‘Hoeken’ (Angles).
This program is on the
.
The Angles program helps you to estimate the sizes of angles.
It’s fun to do this with a friend, and to turn it into a game.
Who is top of the class?
1
1
0
3
2
1
1
2
2
4
3
15 0
0
3
3
13
50 0
12
60 0
ZOETERMEER
110
70
100
80
90
80
100
70
110
60 0
12
4
14 0
0
0
14 0
4
figure 5.25
5
10
170
2
3
20
160
4
5
6
0
15 0
3
Chapter 5
7
160
20
164
You can measure angles using the protractor
printed on your protractor triangle.
At intervals of ten graduation marks, there are
two numbers.
a Which other number is marked at 20?
b Which other numbers are marked at 50, 80
and 120 respectively?
c How can you work out the other number
marked at 40 without looking at the
protractor?
d One of the graduation marks bears only one
number. Why is that?
170
10
O 30
50 0
13
6
7
Theory B
Below, you can see how you can use your protractor triangle to
measure angles.
How to measure angles
90
80
100
70
110
60
120
50
130
30 0
15
0
13 0
5
2
3
150
30
1
17 0
0
140
40
40 0
14
3
ZOET
ERME
ER
2
16 0
0
0
12 0
6
2
11
70 0
10
80 0
1
7
6
5
1
4
3
2
2
160
20
1
0
1
1
2
A
3
170
10
A
4
5
6
7
How many degrees is angle A?
60
120
50
130
0
13 0
5
2
150
30
7
6
5
1
4
3
2
2
3
80
100
70
110
60
120
50
130
30 0
15
2
7
6
5
1
4
3
2
1
0
1
1
2
5
40 0
14
3
ZOET
ERME
ER
2
160
20
1
170
10
160
20
1
0
1
4
90
11
70 0
10
80 0
3
150
30
140
40
3
2
170
10
30 0
15
A
5
40 0
14
3
0
12 0
6
70
110
0
13 0
5
80
100
ZOET
ERME
ER
1
17 0
0
1
17 0
0
140
40
90
2
16 0
0
2
16 0
0
3
4
A
0
12 0
6
4
11
70 0
10
80 0
3
Place your protractor triangle on angle A.
The 0 must be positioned at vertex A, and the
long side of the protractor triangle must be
positioned on top of one of the sides of angle A.
6
6
7
7
From that shank, search along
the scale of the protractor
triangle, marked 10, 20, 30, . . .
When you have arrived at the other side,
read the size of the angle from the scale.
Angle A = 53˚
Instead of writing angle A = 53°, we write ∠ A = 53°.
symbol
The
is not the capital
letter L.
Example
Measure ⬔B in figure 5.26.
30
150
40
140
50
130
20
160
10 0
17
3
140
40
2
1
4
51
7
10 0
16
20 0
3
150
30
1
B
2
Result
130
50
0
120
60
1
110
70
ZO
2
ET
ER
1000
8
ME
3
ER
1
90
4
2
3
60
120
5
7
11 0
0
6
7
B
8
10 0
0
Method
1 Place the 0 of your protractor triangle on B, and the long side
along one of the sides of B.
2 Run your finger along the protractor’s scale marked 10, 20, 30, . . .
3 Continue until you arrive at the other side, then read its position
from the scale.
7
6
⬔B ⫽ 137°
figure 5.26
Lines and angles
165
31 Below is a drawing of four angles, which you can also find in
Write down your
answer as A = ...°.
your
.
a Estimate how wide each of the angles is.
b Measure the size of the angles with your protractor triangle.
Extend the sides if necessary.
c Check your estimates. If you came to within 10° or less, your
answer was right.
A
C
B
D
figure 5.27
T
32 Measure the angles P, Q, R,
S and T. These angles are
.
also in your
Extend the sides if necessary.
P
Q
R
S
figure 5.28
The surveyor
Hans is employed by the Ministry of
Transport, Public Works and Water
Management as a surveyor. ‘I think that
working outdoors is the best part of my job,’
he says. ‘That’s when I go out in the field
with a colleague to measure everything.
We use a theodolite and a long tape
measure.’ A surveyor’s job is technical and
has a lot to do with mathematics. Hans
attended senior secondary technical school,
but you can also go to a higher vocational
college, or study for a technical degree at a
university.
166
Chapter 5
THE OCCUPATION
33 Two lines have been drawn on the photograph
of a starfish in figure 5.29.
Measure the angle between the lines in your
.
figure 5.29
34 Measure angle A on the picture of the temple
of Artemis in your
.
Artemis was the goddess of the hunt
in Greek mythology. She was the
daughter of Zeus, and the twin sister
of Apollo.
The Romans called her Diana.
figure 5.30
35 Draw two suitable lines on the photograph in
, and then measure the angle
your
the tower forms with the ground.
The leaning tower of Pisa
The eight hundred year old leaning tower of
Pisa is once again open to the public.
Thanks to a spectacular rescue operation, the
tower is now standing a little straighter, and
is just as safe as it was three hundred years
ago. However, no more than thirty people
may climb it at any given time.
figure 5.31
Lines and angles
167
36 Carry out the following experiment.
– Look straight ahead and don’t move your
eyes.
– Spread your arms and move them slowly
forward until you can see them both.
– Approximately how wide is your viewing
angle?
A 37
a Plot the points A共1, 0兲, B共⫺1, 1兲 and C共1, 3兲.
b Draw angle A, whose one side passes
through B, and the other through C.
How many degrees is angle A?
c Draw angle C, whose one side passes
through A, and the other through B.
How many degrees is angle C?
A 38
a Plot the points P共1, 1兲, Q共4, 0兲 and
R共0, 6兲.
Draw triangle PQR.
b 䉭PQR has three angles, ⬔ P, ⬔ Q,
and ⬔ R.
Measure all three angles.
Instead of
triangle ABC,
we usually write
ABC.
How did the degree originate?
HISTORY
Sun
Earth
The Earth circles the Sun once a year.
The ancient kingdom of Babylon was at the height of its development between 1900 and
1650 B.C. The Babylonians pressed symbols into clay tablets and then dried the tablets in
the sun. This is how the cuneiform script originated. Thousands of these clay tablets with
cuneiform inscriptions have been discovered. The numbers 60 and 360 in particular appear
to have interested the Babylonians. The number 360 represented the number of days in a
year. Their year was divided into 12 months of 30 days. The Earth circles the Sun once a
year, so they divided this revolution into 360°, i.e. 1° per day.
168
Chapter 5
5.4 How to draw angles
O 39
Fleur wants to divide a cake into 8 equal
, indicate on the
portions. In your
picture how Fleur should cut the cake.
Theory A
The protractor on your protractor triangle can be used
not only for measuring angles, but also for drawing
them.
figure 5.32
How to draw angle A of 130°
110
70
100
80
90
80
100
70
110
60 0
12
ZOETERMEER
50 0
13
4
14 0
0
3
Place the long side of your protractor
triangle along the side of the angle.
Position the 0 on point A.
3
3
15 0
0
0
15 0
3
0
14 0
4
A
12
60 0
160
20
2
170
10
1
6
7
4
5
3
1
2
1
0
1
2
10
170
6
2
20
160
7
2
Draw point A and one side of the angle.
You can place A wherever you like.
13
50 0
1
5
4
3
6
7
12
60 0
110
70
100
80
90
80
100
70
110
60 0
12
ZOETERMEER
50 0
13
4
14 0
0
3
Trace along the scale 10, 20, 30, . . .
until you come to 130°.
Mark it with a dot.
13
50 0
3
0
15 0
3
3
15 0
0
160
20
2
2
170
10
1
3
1
2
1
0
1
2
10
170
4
20
160
5
3
0
14 0
4
A
3
4
5
6
7
A
4
Draw the second side.
A
Lines and angles
169
40 a Draw angle A of 25°.
b Draw angle B of 125°.
c ⬔ C ⫽ 85°. Draw angle C.
d ⬔ D ⫽ 160°. Draw angle D.
41 a Draw angle P of 78°.
b Divide angle P into two angles of equal
size.
42 a Draw angle Q of 168°.
b Divide angle Q into three angles of equal
size.
A 43
The angle of the beam of light from a torch can
be varied between 12° and 44° by turning a
button.
a Draw the smallest angle of the beam in your
.
b Draw the widest angle of the beam as well.
The cake has been divided equally.
figure 5.33
44 Dolphins use echo sounding to find food.
They emit series of clicking noises from their
heads at an angle of 30°. The echoes of these
clicks help them to locate food at distances of
up to 300 m.
A section of the ocean is drawn in your
, on a scale of 1 : 3000.
A dolphin is sending out an echo sounding in the
direction of the arrow.
Which of the fish, A, B, C, D, E or F, can
the dolphin locate from its position?
figure 5.34 Echo sounding is more useful to a dolphin
than the keenest eyesight.
170
Chapter 5
Theory B
How to draw nABC with AB = 3 cm, ∠A = 108° and ∠B = 30°.
100
80
90
80
100
70
110
60 0
12
50 0
13
ZOETERMEER
4
14 0
0
3
0
15 0
3
3
15 0
0
160
20
2
5
2
1
170
10
6
4
3
1
2
1
1
0
10
170
2
3
4
5
6
7
Draw ⬔B of 30°.
The sides intersect at point C.
Mark the intersection of the sides as C.
共don’t erase anything!兲
7
20
160
3
0
14 0
4
110
70
Draw ⬔A of 108°.
B
12
60 0
2
A
3
Draw side AB 3 cms long.
13
50 0
1
A
B
A
B
C
45 Draw a 䉭PQR with PQ ⫽ 6 cm, ⬔P ⫽ 70° and ⬔Q ⫽ 20°.
46 a Plot the points A共⫺2, 1兲 and B共2, 2兲.
Draw the line segment AB.
b Draw 䉭ABC with ⬔A ⫽ 25° and
⬔B ⫽ 110°
c Measure ⬔C.
A line segment has two end points.
A line segment is not infinite.
47 a Plot the points K共⫺3, ⫺1兲 and
M共2, 3兲.
b Draw 䉭KLM with ⬔K ⫽ 44° and ⬔M ⫽ 32°
in such a way, that point L is located below the x-axis.
c Measure ⬔L.
A 48
a Draw 䉭DEF with DE ⫽ 5 cm, ⬔D ⫽ 35° and DF ⫽ 2 cm.
b Measure ⬔E and ⬔F.
49 Window cleaner Helderman places his ladder 2 meters from
the wall.
The ladder is at an angle of 75° to the ground.
a Make a scale drawing, whereby 1 cm is equal to 1 metre.
b Measure the length of the ladder in your drawing.
How long is the ladder in reality?
Lines and angles
171
5.5 Wire models and intersecting lines
O 50
Mr and Mrs De Vries have bought a new family
tent. Mr De Vries wants to know whether the
pipes of the frame are all there. That’s why he
assembles the frame. Look at figure 5.35.
a Is the frame complete?
b Is the frame a cuboid, a pyramid, or a prism?
c How many different kinds of joints are there?
figure 5.35 This is what the frame of the family tent looks
like.
Theory A
Figure 5.36 is a drawing of two prisms.
Most of the faces of a prism are rectangular, however two of them
have a different shape. These two faces are the base and the top.
In figure 5.36, the base and top are coloured.
As you can see, the base is not always on the ground.
top
base
a
b
figure 5.36 The base and top of a prism are always the same shape.
51 The number of faces a prism has depends on the shape of
the base.
a In figure 5.36a, the base is a pentagon.
How many faces does the prism have?
How many vertices and how many edges does it have?
b The base of a different prism is an octagon.
How many faces does this prism have?
How many vertices and edges does it have?
c A prism has eight faces.
Is the base of the prism in the shape of a square, a hexagon,
an octagon or a decagon?
d A prism has twenty vertices.
How many faces does it have?
172
Chapter 5
52 The number of faces a pyramid has is also dependent on the
shape of the base.
a How many faces does a pyramid with a triangular base have?
b How many vertices does a pyramid with a square base have?
c How many edges does a pyramid with a pentagonal base
have?
d A pyramid has seven vertices. How many faces does this
pyramid have?
e How many faces does a pyramid with sixteen edges have?
figure 5.37
Theory B
The photograph on the right shows the wire model of
a cube.
A wire model consists only of edges, that is why you
can see through it.
The frame of the family tent is also a kind of wire
model.
The cube in the photograph was constructed of straws
and string. You could also use cocktail sticks or
matches and Plasticine, or pieces of wire soldered
together. Maybe your school has construction
materials available with which you can easily make
wire models of space shapes.
53 a Make a wire model of a cube.
b Construct a wire model using six rods of
equal length. What is this space shape called?
figure 5.38 A home-made wire model of a cube.
54 Ivo wants to make a wire model of a pyramid
with a pentagonal base.
All the edges are 10 cms long.
How many cms of wire will Ivo need?
Lines and angles
173
55 Sandra, Josse, Maarten and Lisette have a number of sticks, as
shown in the table below. They are going to try to construct
some space shapes with them. However, they must use all of the
sticks.
Sandra
Josse
Maarten
Lisette
4 of 8 cm
8 of 4 cm
6 of 8 cm
6 of 4 cm
9 of 8 cm
18 of 6 cm
If possible, make a sketch to accompany each of
the following questions. If that is impossible,
explain why that is so.
a Can Sandra make a wire model of a cuboid?
b Can Josse make a wire model of a cuboid?
c Can Maarten make a wire model of a prism?
And can he make one of a pyramid?
d Can Lisette construct a prism? Can she make
a pyramid?
O 56
Diana has made a wire model of a cube with
wooden skewers. She then made a drawing of
the wire model, as shown in figure 5.39. You can
easily see where she joined the skewers with
Plasticine.
What is wrong with her drawing?
figure 5.39
Theory C
The cuboid ABCD EFGH is shown three times in figure 5.40.
The green lines BE and BC intersect at point B.
The lines BE and BC are intersecting lines.
The red lines BE and CH have the same direction.
BE and CH are parallel lines.
The blue lines BE and DH are neither parallel, nor do they
intersect. They are called crossing lines.
H
F
E
D
A
D
B
intersecting lines
Chapter 5
A
H
G
F
E
C
figure 5.40
174
H
G
F
E
D
C
B
parallel lines
G
A
C
B
crossing lines
Varieties of line pairs
– intersecting lines have an intersection
– parallel lines have the same direction
– crossing lines are neither parallel, nor do they intersect.
in the workbook.
Use the
H
G
E
F
57 Look at the cube in figure 5.41.
Write down whether the following lines are intersecting, parallel
or crossing.
a AB and BC
d EG and BH
b AC and BE
e BF and DH
c AF and CH
f AF and EG
D
C
A
B
figure 5.41
D
58 Look at the pyramid in figure 5.42.
a Which edge crosses AB?
b Which edges intersect CD?
A
C
B
figure 5.42
E
a Which edges cross DE?
b How many pairs of parallel edges can you find in the
pyramid?
C
D
A
B
figure 5.43
I
A 60
Look at the prism in figure 5.44.
Write down whether the following pairs of lines are intersecting,
parallel or crossing.
a AB and EI
d AF and IJ
b EF and CG
e CE and BH
c GJ and BC
f AC and GI
J
H
G
E
D
F
C
A
B
figure 5.44
Lines and angles
175
Theory D
The faces ABGF and FGHIJ in figure 5.45 intersect.
The cutting edge is line FG.
The faces ABCDE and FGHIJ do not intersect, not even if
you extend them in all directions.
They are parallel faces.
J
I
F
H
G
E
Parallel faces
Parallel faces do not intersect.
D
A
C
The base and top of every prism are parallel.
B
figure 5.45
61 Look at the prism in figure 5.46.
F
a Which lines are parallel?
b Which faces are parallel?
D
E
C
A
B
figure 5.46
T
a Which lines are parallel?
b Are there any parallel faces?
c Can you think of a type of pyramid that has parallel faces?
S
R
P
Q
figure 5.47
A 63
The prism in figure 5.48 has a regular octagonal base.
Which faces are parallel to each other?
N
L
O
F
E
D
P
G
I
C
H
A
figure 5.48
176
Chapter 5
M
B
K
J
A 64
Stefan draws a line through the base of a
cylinder, and another one through the top.
a Are the base and the top parallel?
b Could the two lines be parallel?
Could they intersect, or perhaps cross each
other?
top
base
figure 5.49
A 65
How many pairs of parallel faces can you find in this climbing
frame?
figure 5.50 This climbing frame consists of squares and triangles.
Lines and angles
177
5.6 Summary
§5.1
p. 152
Perpendiculars
In figure 5.51, line p is perpendicular to line q. Line p
is the perpendicular through point A of line q.
p
7
6
k
5
160
20
170
10
150
30
140
40
4
7
0
13 0
5
3
1
60 20
3
1
6
10
0
17
2
100
80
5
20
160
1
R
0
30
150
80
100
TERMEE
90
ZOE
4
3
2
70
110
1
1
40
140
1
60
120
2
2
1
40 0
14
3
1
70
11
0
2
4
4
1
5 1 70
0
5
6
8
10 0
0
50
130
3
10
170
3
2
3
2
16 0
0
2
1
50
130
60
120
0
15 30
0
l
2
11
70 0
q
3
16
0
20
150
30
6
7
140
40
130
50
120
60
110
70
0
10
80
90
90
7
figure 5.51 p is perpendicular to q.
figure 5.52 k and l are parallel lines.
Parallel lines
Parallel lines do not intersect, not even if you extend them.
§5.2
p. 158
Angles
The figure on the right shows angle A.
Every angle consists of a vertex and two sides.
e
sid
A
figure 5.53 No matter how far you
extend the sides, the angle remains the
same size.
An acute angle is smaller than a right angle.
A wide angle is larger than a right angle.
right
acute
side
wide
flat
13
50 0
110
70
12
0
60
100
80
90
90
80
100
70
110
60
0
12
50 0
13
3
3
0
15
30
2
2
160
20
1
1
170
10
5
4
3
2
1
0
1
2
3
4
5
6
7
figure 5.55 ⬔P = 153°.
6
P
7
10
170
Chapter 5
20
160
178
30
15
0
Angles are measured in degrees.
A right angle is 90°, a flat angle is 180°.
Use the protractor on your protractor triangle to measure
angle P on the right.
⬔P ⫽ 153°.
40
p. 162
0
How to measure angles
14
§5.3
0
14
40
figure 5.54 A right angle, an acute angle, a wide angle and a flat angle.
§5.4
How to draw angles
p. 169
Use the protractor on your protractor triangle to draw angles.
The figure below illustrates step-by-step how to draw angle
Q if ⬔Q ⫽ 73°.
13
50 0
110
70
12
0
60
100
80
90
90
80
100
70
110
60
0
12
3
2
2
50 0
13
0
15
30
30
15
0
3
4
14 0
0
0
14 0
4
73°
160
20
20
160
1
170
10
7
6
5
4
3
2
1
1
0
1
2
10
170
3
4
5
6
7
Q
73°
Q
Q
figure 5.56 ⬔Q = 73°.
p. 171
How to draw triangles
Tackle the following exercise
C
Draw 䉭ABC with AB ⫽ 4 cm,
⬔A ⫽ 120° and ⬔B ⫽ 20°.
120°
as follows:
1 Draw line AB ⫽ 4 cm.
2 Draw ⬔A ⫽ 120°.
3 Draw ⬔B ⫽ 20°.
4 Mark the intersection as C.
§5.5
p. 172
B
figure 5.57
Intersecting, parallel and crossing lines
There are three possibilities for the spatial
positioning of two lines.
– Intersecting lines have an intersection.
Look at the green lines.
– Parallel lines have the same direction.
Look at the red lines.
– Crossing lines are not parallel, neither do
they intersect.
Look at the blue lines.
20°
4 cm
A
H
G
E
F
D
C
p. 176
Parallel faces
In figure 5.58, ABCD and EFGH are
parallel faces because they do not intersect.
A
B
figure 5.58
Lines and angles
179
Diagnostic test
.
test on the
§5.1
1
Look at figure 5.59.
a Which of the lines are perpendicular to
1
line a?
b Are there any other lines that are
perpendicular to each other? If so,
which ones?
c Which lines are parallel to each other?
a
b
c
revision exercise
d
e
f
2
a Plot the points A共6, 0兲, B共⫺1, 4兲
g
and C共4, 4兲. Draw line l through
2
points A and B.
figure 5.59
b Draw line m through point C,
perpendicular to line l.
c Draw line n through point B, perpendicular to line l.
d Draw line q through point O, parallel with line l.
revision exercise
§5.2
3
a How many right angles can you see in figure 5.60?
b How many wide angles are there?
3, 4 c How many acute angles are there?
G
D
H
C
F
revision exercises
A
E
B
figure 5.60
§5.3
4
In figure 5.61, four angles have been drawn with an arc.
. Write
Measure these four angles in your
5, 6 your answers as follows: ⬔P ⫽ . . ., ⬔Q ⫽ . . ., etc.
S
R
revision exercises
5
How many degrees is the angle of the clock hands at
revision exercise a five o’clock
7 b half past four
c a quarter past five?
180
Chapter 5
T
P
Q
figure 5.61
D
§5.4
6
revision exercise 8
a Draw angle A of 108°.
b Divide angle A into three angles of equal size.
a Draw 䉭PQR with PQ ⫽ 4 cm, ⬔P ⫽ 30° and
⬔Q ⫽ 118°.
revision exercise 9
b Measure ⬔R.
7
8
Figure 5.62 shows a partially opened
revision exercise bridge. This drawing can also be found in
. 1 cm in the drawing is
10 your
equal to 1 meter in reality.
a Measure ⬔S.
b Measure how many dm there are
between point T and the water level.
When the bridge is fully raised,
⬔S ⫽ 83°.
c To show this, draw ⬔S using points T
and P, on a scale of 1 : 100.
9
The four cubes shown below each have
revision exercises two red lines drawn on them.
11, 12 For each cube, write down whether the
lines are intersecting, parallel or crossing.
§5.5
a
figure 5.62 The bridge over the canal.
b
c
d
figure 5.63
10 Femke has made a wire model of a space shape from
revision exercise
13
6 sticks of equal length.
a Make a sketch of her space shape. What is it called?
b Does this shape contain parallel edges?
c How many pairs of crossing edges can Femke see?
Lines and angles
181
Revision
§5.1
1
You can check whether two lines are perpendicular or parallel by
using your protractor triangle. See figure 5.64.
p
k
7
6
7
l
6
5
20
160
a
150
30
140
40
0
13 0
5
2
3
30
150
40
140
2
100
80
1
1
1
60
120
R
80
100
3
0
2
70
110
1
1
70
11
0
3
60
120
2
3
2
50
130
1
5 1 70
0
1
6
3
40 0
14
7
2
3
16
0
20
150
30
140
40
130
50
120
60
110
70
0
10
80
90
90
e
4
2
16 0
0
2
8
10 0
0
90
ZOE
TERMEE
0
50
130
1
11
70 0
2
1
4
15 30
0
10
0
17
160
20
170
10
4
4
3
3
1
60 20
q
5
5
10
170
6
7
b
figure 5.64
figure 5.65
a Which of the lines in figure 5.65 are perpendicular to line d?
b Which of the lines are parallel?
§5.2
2
a Plot the points A共⫺2, 1兲, B共3, 2兲, C共1, ⫺1兲 and S共1, 4兲.
Draw line l through points A and B, and line m through
points A and C.
b Draw line p through S, perpendicular to line l.
c Draw line q through S, perpendicular to line m.
d Draw line r through C, parallel with line l.
e Draw line s through S, parallel with line l.
3
In the rectangle on the right, five angles are
marked with arcs.
a Which of those angles are right angles?
b ⬔D is acute, because . . .
c Which other angle is also acute?
d Which type of angle is ⬔E?
figure 5.66
4
182
Fill in larger than, smaller than or equal to.
a An acute angle is . . . 90°.
b A wide angle is . . . 90°.
c A right angle is . . . 90°.
d A flat angle is . . . 180°.
Chapter 5
c
d
70
110
0
14
40
3
150
30
160
20
3
50
0
13
40 0
14
2
1
1
4
60
120
3
2
170
10
7
80
100
10
170
5
90
90
20
16
0
2
1
1
0
2
3
4
6
5
7
A
6
100
80
3
15 0
0
To measure angle A, place the 0 of your
protractor triangle on point A. Turn the
protractor triangle until the long side is situated
on top of one of the sides of angle A.
As you can see, ⬔A ⫽ 70°.
Measure the angles drawn in figure 5.68 in your
.
Write down your answers as follows:
⬔B ⫽ . . .°, ⬔C ⫽ . . .°, etc.
11
0
70
5
12
60 0
§5.3
0
13 0
5
R
figure 5.67
E
C
B
D
figure 5.68
6
Measure the three angles of triangle PQR in your
.
R
P
Q
figure 5.69
7
a How many degrees does the small hand of a clock advance in
one hour, in half an hour, and in a quarter of an hour?
b How many degrees does the large hand of a clock advance in a
quarter of an hour, in 10 minutes, and in 5 minutes?
c How many degrees does the angle of the hands measure at
six o’clock, at a quarter past six, and at five minutes past six?
Lines and angles
183
100
80
90
90
80
100
70
110
60
0
12
3
3
2
2
1
1
50 0
13
30
15
0
0
15
30
0
14 0
4
110
70
4
14 0
0
7
6
5
4
3
2
1
0
1
2
10
170
170
10
160
20
20
160
3
4
5
6
7
9
To draw angle A of 105°, first draw point A
and one side of the angle.
Use the protractor on your protractor triangle to
mark 105° with a dot. See figure 5.70.
a Draw angle B of 135°.
b Draw angle C of 82°.
c Divide angle C into two angles of equal
size.
12
0
60
8
13
50 0
§5.4
figure 5.70
In triangle ABC, ⬔A ⫽ 100°, AB ⫽ 6 cm and ⬔B ⫽ 32°.
a First draw the line segment AB measuring 6 cm, and add
the letters.
b Side AB is one side of ⬔A.
Draw the other side of ⬔A.
c Also draw the other side of ⬔B. Don’t forget that ⬔B is
acute.
d Complete the triangle.
10 In figure 5.71a, you can see a closed level
crossing. Figure 5.71b shows the same level
crossing, but here the barrier is raised to an
angle of 30°. The segment ST is 5 meters long.
In figure 5.71b, you can check by measuring that
point T is 3.5 cms above the ground. In reality,
T is 3.5 ms above ground level.
, draw the situation
a In your
where ⬔S ⫽ 70°.
b Measure how many dm point T is now
above ground level.
figure 5.71 A level crossing barrier at an angle of 0° and
30°. The scale is 1 : 100.
§5.5
11 In figure 5.72 on the next page, you can see a drawing of the
roof of a house with a dormer window, which has ten red lines
marked on it.
For questions a, b, and c, you can choose the following answers:
they intersect, they are parallel or they cross each other.
184
Chapter 5
R
figure 5.72 Two lines intersect, cross each other, or are parallel.
a Gutter e and the edge of the roof d intersect. What do you
now know about e and d?
b Gutter e and the ridge of the roof f go in the same
direction. What do you now know about lines e and f?
c The side of the house i and ridge f are definitely not
parallel. To intersect, they must have an intersection! Is there
an intersection? What do you know about lines i and f?
d Write down all the lines that are parallel to b.
e Write down all the lines that intersect b.
f Write down all the lines that cross b.
12 Figure 5.73 is a wire model of the pyramid
TABCD.
To make the model more stable, wires AC, BD
and ST have been added.
a Which wires intersect AB?
b Which wires are parallel to AB?
c Which wires cross AB?
d How many wires intersect ST?
figure 5.73 The base of this pyramid is a square.
13 a What is the least number of sticks with which
you can make a wire model of a prism?
b Make a sketch of the prism and add letters at the vertices.
c Write down four pairs of crossing edges.
d Write down which edges are parallel.
Lines and angles
185
Extra material
Arabic numerals
1
The numerals 1, 2, 3, 4, 5, 6, 7, 8 and 9 were first used by
Phoenician traders. During the Arab civilisation, which lasted
from about 750 to 1250 A.D., they were introduced throughout
almost the entire world. This is why they are called Arabic
numerals. In their earliest known form, the numerals looked like
this.
figure 5.74
You can discover a pattern in these symbols by
looking at the number of angles they contain.
On the right, the angles of the numerals 1 and 4
have been marked with arcs.
a Explain why the very first numeral is entirely
round.
b Check whether the number of angles is
correct for each numeral.
figure 5.75
Tall letters
2
Figure 5.76 is a sentence.
a You have to look at the book in a
particular way to be able to read this
sentence.
What does the sentence say?
b Write a message in the same way and
ask someone else to read it.
figure 5.76
186
Chapter 5
E
Course
0
33
0
340
350 360/0 10
NNE
31
0
NNW
NW
30
WNW
W
80
WSW
SE
SW
13
SSE
0
14
0
15
0
S
0
23
0
24
0
ESE
250
90 100
110
12
E
SSW
0
0
170 180 190 20
0 2
160
1
22
4
At how many degrees are the following courses
marked?
a NE
b NW
c SE
d SSE
e SSW
f WSW
70
ENE
3
60
NE
260 270 280 29
0
30
40
32
0
20
50
A compass is often used in aviation and navigation.
The compass in figure 5.77 shows 16 compass points.
A compass with only 16 points is not very accurate.
A compass is therefore also divided into degrees.
The scale begins at North 0°. East is at 90°, and West
is at 270°.
If a ship is sailing in a south-westerly direction, its
course is 225°.
figure 5.77
Kevin walks from A to B. Below, you can see how you can
measure his course.
Plotting a course from A to B
2
1
N
A
A
B
B
Draw a N-S line through point A.
Measure ⬔A indicated by the
arrow, and then add 180°.
How many degrees is the course from A to B?
Lines and angles
187
5
a Floris Jan flies from Rotterdam airport to
Eindhoven airport. Use the map in figure 5.78
to measure his course 共in degrees兲.
This map can also be found in your
b Antoinette flies from Zeeland airport to Texel.
How many degrees is her course?
c Jasper flies from Ameland to Seppe. How
many degrees is his course?
Ameland
Eelde
Texel
.
Hoogeveen
Emmeloord
Teuge
Schiphol
Twente
Hilversum
Rotterdam
6
a
Antoinette takes off from
Teuge airport. Her heading is 34°. After she
has flown 62 kms, she lands. Which airport
has she landed on? Use the scale shown on
the map.
b Floris Jan takes off from Eindhoven airport.
His course is 345° and he has to fly 190 kms.
Where is he heading?
Zeeland
Seppe
Eindhoven
0
Z-Limburg
25 50 75 100 km
figure 5.78 Private airfields.
M6
M
A5
50
100 150km
BIRMINGHAM
FISHGUARD
A17
NORWICH
7
M4
M40
A11
A ferry sails from Harwich to Hoek van
CARDIFF
A45
HARWICH
OXFORD
M11
AMSTE
Holland. The crossing is indicated in figure 5.79
A12
M5
M4
LONDON
M2
A30
with an arrow.
SOUTHAMPTON M3
A36
HOEK VAN HOLLAND
M20
A35
DOVER
M23
OOSTENDE
ROTTER
PLYMOUTH
From the direction of the arrow, you can see that
PORTSMOUTH
NEWHAVEN
ANTWERP
BRUGES
CALAIS
the course is approximately 110°. By measuring
BRUXELLE
E3
LILLE
the length of the arrow, you can work out the
CHERBOURG
A2
DIEPPE
CHARL
LE HAVRE
distance of the crossing.
AMIENS
E411
REST
ROUEN
N13
N12
The arrow is 1.7 cms long, therefore the crossing
figure 5.79 This map only shows the ferries of Stena
is 1.7 ⫻ 100 ⫽ 170 kms.
How many kms is the distance Sealink.
a
from Newhaven to Dieppe, and how many
degrees is the course?
b How many degrees is the course from Dieppe
to Newhaven?
c What is the connection between the courses
in questions a and b?
8
Look at figure 5.80. A Channel Islands mail
plane makes a round trip from Jersey via
Guernsey and Alderney.
a What is the course and distance from Jersey
to Guernsey?
b And from Guernsey to Alderney?
c And from Alderney to Jersey?
188
Chapter 5
figure 5.80 Although the Channel Islands are close
to France, they belong to England.
E
9
A city is advertising an industrial area that is to
be developed.
Figure 5.81 shows the advertisement.
Indicate the location of the new industrial area
.
on the map in your
10 Schiphol has four runways. The Buitenveldert
There are plans to add a fifth runway to
Schiphol.
Imagine that this runway will run parallel to and
south of the Kaagbaan runway.
e Which numbers and letters would be used to
mark the new runway?
figure 5.81 This is the advertisement for the new industrial
area.
360˚
0˚
Zwanenburg
Amsterdam
19 R
rgbaan
Badhoevedorp
19 L
09
270˚
01 L
aan
agb
Hoofddorp
Ka
06
Amstelveen
27
rbaan
Buitenveldertbaan
Aalsmee
Zwanenbu
runway lies in an East-West direction, i.e. from
90° to 270°. It is called the 09-27 runway, as the
last zero is dropped.
At the point on the compass that reads 90°, the
Buitenveldert runway is marked with the
number 27. It would appear to be more logical if
it were marked 09. However, it’s the other way
around because the flight path, the direction of
landing or take-off, is used.
a An aeroplane approaches Schiphol from the
South-West. What course does the pilot steer
to land?
b Calculate the angle between the Kaagbaan
and Buitenveldert runways.
You are not allowed to measure it!
c What have you noticed about the difference
between the numbers for each runway?
Give an explanation.
d The Zwanenburg and Aalsmeerbaan runways
are also marked with the letters L and R.
Explain why this is necessary.
Why isn’t the Zwanenburg runway marked
with two Ls?
90˚
24
01 R
Aalsmeer
180˚
figure 5.82
Lines and angles
189
chapter
IMA
Mathematics and art
The photograph shows an unusual work of art by George W. Hart.
This American mathematician and artist makes beautiful space
shapes using common materials such as paper, toothpicks,
pipe-cleaners and lollypop sticks. If you would like to know
more about his work, surf to
http://www.georgehart.com/.
• Which shape can you recognise in the mobile pictured in the
facing photograph?
• What is it made of?
• How do you think George W. Hart constructed this space
shape?
190
IMA
what you can learn
ICT
– From the examples shown in this IMA (Integrated
Mathematical Activities) section, you will discover
that a great deal of mathematics can be found in
art.
– In a woodcut by M.C. Escher, you will come across
ants in a never-ending march along a mysterious
loop.
– You will learn how to design patterned surfaces.
– At the end of the IMA, you will integrate your
answers and experiences in a report.
– Applet ‘Tegelpatronen’ (tile patterns) on page 196.
The Möbius strip
1
The figure on the right shows a 1963 woodcut
by M.C. Escher.
a How many ants does it contain?
b One of the ants is running faster than the
others.
How many ants does it pass?
figure IMA.1 ‘Möbius Strip II’ by M.C. Escher.
2
a Take a sheet of A4 paper.
Cut a lengthways strip 3 cm wide from it.
Draw the letters A and B on the strip,
exactly as shown in figure IMA.2a.
Grasp one end of the strip in one hand, and
twist the other end by half a turn with the
other.
Glue the ends together, so that A is on top
of A, and B is on top of B, as in figure
IMA2.b.
b Mathilde says the strip has only one edge.
Do you think she is right?
c Mathilde also says that this strip doesn’t have
in inside as opposed to an outside, because
there is only one side.
Do you agree with Mathilde?
d What does your strip have in common with
the one shown in figure IMA.1?
figure IMA.2 Twist by a half-turn, and glue A to A and
B to B.
The Möbius strip
The Möbius strip has one side and one edge.
August Ferdinand Möbius
August Ferdinand Möbius (1790 – 1868) was a German
mathematician and astronomer. He was director of the Leipzig
observatory and professor of astronomy.
The Möbius strip (1858) was an important contribution to the
development of topology, which is the part of mathematics
concerned with the deformation of geometrical figures.
192
IMA
HISTORY
3
Take the Möbius strip you made in exercise 2, and cut it
lengthways right down the middle.
Is the result still a Möbius strip? You can check this by tracing
around the strip with a pen.
4
a Make a new Möbius strip. Slit it lengthways 13 from the edge
all the way around. Describe the result accurately.
figure IMA.3 It is not easy to predict what the result will be if you slit a Möbius strip
lengthways 13 from the edge.
b Slit a Möbius strip lengthways 14 from the edge all the way
around. Describe the result.
c Carefully describe what happens when you slit a Möbius strip
lengthways 15 from the edge.
5
6
a Cut two 2 cm wide strips from an A4 sheet of paper, and one
2 cm wide strip from a coloured sheet. Number the ends of the
one white strip 1 and 2 respectively, the ends of the coloured
strip 3 and 4, and the ends of the second white strip 5 and 6.
b Place the ends numbered 1, 3 and 5 on top of each other.
The coloured strip is now in the middle. Hold these three ends
together, and give the other three ends a half-twist. Glue end 6
to end 1, end 4 to end 3, and finally, end 2 to end 5.
c Carefully separate the three strips and describe the result.
figure IMA.4
a Cut two 2 cm wide strips lengthways from an A4 sheet of
paper. Make a Möbius strip with the one, and a regular loop
with the other.
b Glue the strips together as shown in figure IMA.5. Slit both
strips lengthways down the middle. Describe the result.
figure IMA.5
Mathematics and art
193
Patterned surfaces
Everybody knows what a tiled floor looks like.
A lot of identical tiles are laid adjacent to each other,
similar to the pieces in a puzzle, so that they cover the
entire floor. Sometimes tiles of different sizes are
used. You can not only discover a regular repetitive
pattern in floors that are tiled like this, but for instance
also in rugs, curtains or wallpaper.
The smallest repeating part is called a motif.
An area filled with motifs is called a 共regularly兲
patterned surface.
The tiled floor shown on the right consists of
octagonal and square tiles. Two possible motifs are
shown below.
Can you think of another possible motif for this floor?
figure IMA.6
7
The Dutch graphic artist M.C. Escher 共1898⫺1972兲 is
world-famous. Much of his graphic art is optically surprising.
Below are two of his drawings consisting of patterned surfaces,
.
which can also be found in your
figure IMA.7
figure IMA.8
a Name a number of differences between the two drawings.
b Each of the drawings has a motif. Colour them in your
.
194
IMA
How to make a patterned surface
It is boring to decorate a surface using only squares. However, by
systematically deforming the squares, you can achieve some very
pleasing results. Figure IMA.9 shows how this can be done.
figure IMA.9 Four steps to turn a boring square into a pleasing motif.
figure IMA.10 A surprising patterned surface.
8
Figure IMA.11 shows how a new motif is created using an
equilateral triangle with sides measuring 3 cm. An equilateral
triangle with sides measuring 1 cm is cut out from the middle of
one of the sides, and added on to one of the opposite sides.
figure IMA.11 A new motif in two steps.
a Copy the motif on a loose sheet of paper and cut it out six
times.
a Make a patterned surface with this motif, and colour it in
attractive colours.
c Design your own motif, beginning with the triangle in figure
IMA.11. Use it to make a patterned surface.
Mathematics and art
195
in the workbook.
Use the
9
The figure below shows a working diagram for designing patterned surfaces.
.
Create an attractive patterned surface in your
Working diagram
1 Use one of the pages in your
.
A
C
2 Choose two points and join them with a
graceful line.
See figure IMA.12.
3 Transfer your line to two other points.
See figure IMA.13.
C
B
D
figure IMA.13
A
4 Join points A and C in a different way,
and then transfer this joining line to points
B and D.
See figure IMA.14.
You have now made a motif.
196
C
figure IMA.14
5 Complete the patterned surface.
See figure IMA.15.
figure IMA.15
6 Make the patterned surface more
attractive by adding colour, eyes, etc.
figure IMA.16
IMA
D
figure IMA.12
A
B
D
B
Inpossible shapes
Escher became famous, amongst others, for his drawings titled
‘Belvedere’ and ‘Waterfall’.
figure IMA.17 Belvedere
figure IMA.18 Waterfall
10 a What do you notice about the Belvedere drawing?
b What do you notice about the Waterfall drawing?
11 Not all so-called impossible shapes are as complex as the two
drawings by Escher. Look at the ‘cube’ the man sitting on the
bench next to the Belvedere is holding. See also the figure on
the right.
Create your own drawing of an impossible shape. Add striking
colours.
12 Create your own work of art using everyday objects, for
example plastic spoons, plastic cups, knives and forks, straws,
pens or used diskettes.
figure IMA.19
13 See if you can find any works of art based on mathematics in
the area where you live. You could, for instance, visit a museum,
interview an artist, or go for a walk where there are buildings
with unusual architecture.
Give a short description of what you have found. Take some
digital photographs and include them in your report.
Mathematics and art
197
Combined Exercises
1 Space shapes
1
A group of rabbits have congregated in a
meadow. Suzanne and Nicole are watching
them, but because there are bushes in the way,
they can’t see all of them.
a From whose position can the most rabbits be
observed?
b Which of the rabbits can they both see?
c Are there also some rabbits that neither of
them can see? If so, which?
figure C.1
2
The photograph below shows five different geometrical space
shapes. The figure on the right shows a map of their positions.
a Write down the name of each of the shapes.
b How many curved faces do the shapes have altogether?
And how many flat faces?
figure C.2 These objects are situated on a dyke near
Gorinchem. They form a work of art called “Geometry in a
landscape”, designed by R. Winiarski.
figure C.3 A bird’s-eye view of the sculpture.
c Which space shape is located at position D?
And which at position A?
, mark the position where the
d In your
photographer must have stood.
198
Combined Exercises
3
The largest church in the smallest country in the world is
St. Peter’s 共Basilica di S. Pietro兲, which is situated in Vatican
City in the heart of Rome. The church square 共Piazza S. Pietro兲
is bordered by two semi-circular ranks of columns. An obelisk
rises from the centre of the square. There are five photographs of
, which have been taken from
St. Peter’s in your
points A, B, C, D and E.
Write down next to each photograph the position
from which it was taken.
4
Merel has a bag containing 120 cubes whose
edges are 1 cm long, which she uses to build a
cuboid. Calculate what the dimensions of the
cuboid are if its height is
a 5 cm
b 4 cm
figure C.4
5
Look at figure C.5.
a How many radii and diameters are there?
b Draw quadrangle ABCE in your
.
What is the name of quadrangle ABCE?
D
E
C
6
Bart has a box measuring 20 by 18 by 24 cm
containing building blocks.
The building blocks are also cuboids measuring
3 by 4 by 5 cm.
How many building blocks fit into the box?
A
B
7
a Draw rectangle ABCD, with AB ⫽ 4 cm
and BD ⫽ 2 cm.
figure C.5
b Side AB is also the side of square ABEF.
Draw the square in such a way, that the
rectangle and the square are not superimposed
on each other.
c Draw diagonal AC in the rectangle.
d AC is one side of square ACGH.
Draw this square in such a way, that it is not
superimposed on square ABEF.
e Draw CF. Of which rectangle is CF the
diagonal?
Combined Exercises
199
8
The base of a cuboid is 4 by 3 cm, and its height is 5 cm.
a Draw a net of the cuboid.
b The bottom half of the cuboid is blue.
Colour the relevant parts of the net blue as well.
c On the top of the cuboid a circle has been drawn which is as
large as possible, and whose centre is exactly in the centre of
the cuboid.
Draw the circle in the net.
9
Look at the cuboid in figure C.6.
a Which faces have the largest surface area?
How large is that area?
, draw diagonal HF
b In your
across the top, and diagonal DB across the
base.
c What is the shape of quadrangle DBFH?
d Is the surface area of DBFH larger or
smaller than that of ABFE? Explain your
answer.
H
G
E
F
D
C
3 cm
A
6 cm
figure C.6
10 The nets drawn below are of three different space shapes.
Their dimensions are marked in cm.
3
5
6
7
8
a
b
13
c
figure C.7
a Which type of space shape does each net represent?
b Ben has drawn the nets to scale and has assembled the space
shapes. He has positioned them in such a way, that the tallest
side is up.
Which of the space shapes is the tallest?
c Ben places the space shapes on top of each other.
Figure C.8 shows the top view.
How tall is his tower?
figure C.8
200
Combined Exercises
2 cm
B
2 Numbers
11 Look at the advertisement on the right.
a Mrs Ligtvoet has bought a lamp at HUISKENS
that would normally cost € 190.
How much did she have to pay for it now?
b Margot has bought a desk lamp for € 18.
What did it cost originally?
c On one of the Saturdays during the sale,
HUISKENS sold goods to the value of € 728.
How many Euros discount did HUISKENS give
away on that day?
figure C.9
12 Roeland is not called ‘Speedy Roeland Gonzales’ for nothing.
Just look at the way he has done his fraction exercises. In his
haste, he forgot to write down the plus, minus, multiplication and
division symbols. Copy the sums and fill in the correct symbols.
a 5 . . . 12 ⫽ 5 12
d
1
2
... 2 ⫽ 1
g
1
2
. . . 14 ⫽ 14
b 5 . . . 12 ⫽ 4 12
e
1
2
. . . 2 ⫽ 2 12
h
2
3
. . . 2 ⫽ 13
c 5 . . . 12 ⫽ 2 12
f
1
2
. . . 2 ⫽ 14
1
i 10 . . . 10
⫽1
13 Calculate the following sums without using your calculator, and
write down the step in-between.
a
3
8
⫹ 25
d
b
3
4
1
⫺ 10
e 1 34 ⫺ 1 13
h 8 27 ⫺ 3 13
f 800 ⫺ 2 13
i 2 13 ⫻ 24
c 1 34 ⫻ 56
3
4
⫻ 480
g 6 13 ⫻ 1 12
14 Read the article on the right and answer the
following questions:
a How many cm of spaghetti are there in one
packet? How many km does that add up to?
b How many packets of spaghetti do the Dutch
eat in one year?
c How many meters of spaghetti does a Dutch
person eat in one year?
How many packets does that add up to?
Netherlands spaghetti-land?
The Dutch love to eat spaghetti.
The 16 million inhabitants of the Netherlands
annually consume more than 3 million
kilometres of spaghetti, and they do this
centimetre by centimetre, for each string of
spaghetti is 24.5 centimetres long. Each
packet contains 625 strings of spaghetti.
Combined Exercises
201
15 WAMMES petting zoo houses 72 animals.
The figure below shows the groups according to type of animal.
1
2
part
poultry
...
1
3
part
animals
72
...part
...part
ducks
...
1
12
part
horses
...
...part
goats
8
...part
cows
...
quadrupeds
...
1
4
part
...part
hens
...
...part
cockerels
1
1
3
Flemish Giants
...
chickens
...
part
rabbits
...
...part
figure C.10
Fill in the blanks in your
part
bucks
...
...part
does
...
1
2
miniature rabbits
...
. Sometimes a fraction is required, and sometimes a number.
16 Mr Van der Heyden departs on a long car journey with a full tank of petrol.
After he has driven about three-quarters of the way, the petrol tank still appears
to be one-third full. Will Mr Van der Heyden still have to stop for petrol?
17 Calculate the following without using your calculator.
a 共17 ⫺ 8兲 ⬊ 3 ⫹ 8
b 18 ⫹ 93 ⫺ 48 ⬊ 4 ⫻ 共17 ⫺ 12兲
c 150 ⫺ 共8 ⫺ 3兲 ⫻ 8 ⬊ 10
d 93 ⫺ 8 ⫻ 共7 ⫺ 7兲 ⫹ 共3 ⫹ 11兲 ⬊ 7
e 80 ⬊ 40 ⫻ 2 ⫺ 共7 ⫺ 6兲 ⫻ 3
f
1
2
⫹ 13 ⫻ 14 ⫺ 15
18 DE GROENE DWERG 共The Green Dwarf兲, manufacturer of
washing-powder, produces 6,000 kg of washing-powder
a day. Each packet contains 2 12 kg of washing-powder.
a How many packets of washing powder does
DE GROENE DWERG produce each day?
b Estimate how many tons of washing-powder
DE GROENE DWERG produces annually.
19 There are approximately five hundred billion worms
living in Dutch soil. They are about 10 cm in length. If
you were to lay all the worms end to end, they would
measure 1,250 times the circumference of the Earth.
Use this information to calculate the Earth’s circumference.
202
Combined Exercises
1 ton = 1000 kg
20 a Read the newspaper article on the right.
b How many kg of cat feed does a cat consume
in 15 years?
c Calculate the price of a kilogram of cat feed.
d Dogs live approximately 12.5 years.
What does it cost to feed a dog each month?
e The article mentions that there are half a
billion pets in the world.
Calculate whether this statement is true.
Each dog consumes a 3,000 of
dog feed
The average cat that lives to be 15 years old
and consumes 1.6 kilograms of cat feed
weekly, is worth € 1,750 in turnover during
its lifetime. A medium-sized dog consumes
about € 3,000.
There are half a billion pets in the world.
Fish head the list, with 178 million being
kept domestically. Cats come in second
共98 million兲. There are 92 million dogs,
88 million birds, and 28 million other small
mammals being kept as pets.
21 Read the newspaper article about mobile
telephones. Using the information given in the
article, fill in the table below. Enter the numbers
in millions to one decimal place.
MOBILE TELEPHONES IN THE
NETHERLANDS
number of
subscribers
second
quarter
third
quarter
KPN
O2
Rest (Ben,
Dutchtone
and Telfort)
...
...
...
...
...
...
total
...
5.3 million
More than 5 million Dutch
phone mobile
During the third quarter of 1999, 5.3 million
Dutch people were using mobile phones.
This is an increase of no fewer than 800,000
compared to the previous quarter.
KPN is the leader with a three-fifths share of
the market. KPN was able to welcome
400,000 new subscribers during the last
quarter. O 2 had a three-tenths share in the
third quarter, and gained an increase of
0.2 million subscribers.
The rest of the mobile market is shared by
Telfort, Ben and Dutchtone.
22 Flower merchant Arie de Reus is once again offering a special
deal. Begonias, which are normally sold at four for € 13, are now
going at seven for € 25.
a Draw up a relationship table for both prices, making sure that
you end up with the same number of begonias in both tables.
b Are you impressed with Arie’s special deal? Why 共not兲?
23 Round off the number 135.595996 to:
a integers
b one decimal place
c two decimal places
d three decimal places
e four decimal places
f five decimal places
Combined Exercises
203
3 Locating points
E41
W40th ST
E40
W39th ST
E39
W38th ST
E38
W33rd ST
W26th ST
W25th ST
W24th ST
Y
WA
AD
W28th ST
O
BR
W29th ST
W27th ST
e Where in New York is the dividing line
between East and West?
EMPIRE STATE
BUILDING
b Place the following numbers in their correct place, adding a
stroke on the number line for each of them.
⫺3.3
⫺0.4
3.5
3.9
0.8
⫺3 34
26 Fill in ⬎ or ⬍ or ⫽ where appropriate.
204
a 3 . . . ⫺4
e ⫺5 78 . . . ⫺5.7
b ⫺4 . . . ⫺5
f ⫺3 23 . . . ⫺3.66
c ⫺4.2 . . . ⫺4.3
g ⫺0.125 . . . ⫺ 18
d ⫺5.4 . . . ⫺5.5
h ⫺1.2 . . . 0
Combined Exercises
E36
E35
E 34th STREET
E33rd ST
E32nd ST
E31th ST
E30th ST
E29th ST
E28th ST
E27th ST
E26th ST
E25th ST
E24th ST
E 23rd ST
W22nd ST
25 a Draw a number line numbered from ⫺6 to 5.
E37
MADISON
SQUARE
PARK
W 23rd ST
figure C.11
PARK AVENUE
BRYANT
PARK
FIFTH AVENUE
AY
ADW
PUBLIC LIBRARY
W36th ST
W31th ST
W30th ST
GRAND
CENTRAL
STATION
W43rd ST
W37th ST
W32nd ST
E45
W44th ST
W 34th ST
John
.
BRO
AVENUE
TIMES
SQR
AVENUE OF AMERICAS
W 42nd ST
SIXTH AVENUE
Start at the red arrow
W45th ST
SEVENTH
see the map of a section of New York.
Many street names consist of numbers, such as
Fifth Avenue and E23rd Street 共23rd Street East兲.
a What strikes you about the horizontal street
names on the map? What is special about the
vertical street names?
b Name an advantage and a disadvantage of
having numbers for street names.
c John lives on W29th Street. He wants to get to
Grand Central Station via 7th Avenue. At
which street will he have to turn right to get
there?
d A group of tourists are walking from the
Empire State Building to Grand Central
Station. Which streets do they walk along?
MADISON AVENUE
24 John lives in New York. On the right, you can
27 This exercise is about the following points: A共3, 6兲, B共⫺2, 3 12 兲,
C共0, ⫺3兲, D共⫺2, 2 12 兲, E共4, 0兲, F共⫺2 12 , ⫺2兲, G共2 12 , ⫺2兲,
H共4, 4兲 and I共2, 3 12 兲.
a Which point has 2 12 as its y-coordinate?
b Which points have the same x-coordinate?
c Which point is located on the y-axis?
d Which points are not grid points?
e Where do you arrive if you proceed 3 places to the right, and
9 places up from C?
f Where do you arrive if you proceed 2 12 places to the right, and
2 places up from F?
g Describe how you would proceed to get from D to G.
h Point P is 5 places to the left, and 3 12 places below B.
Write down the coordinates of P.
28 Below is a series of increasingly large triangles.
y
E
6
5
D
4
C
4
3
B
2
O
2
A
1
1
2
3
4
5
6
7
8
10
12
14
16
18
20
x
figure C.12
a Copy the table below and fill in the blanks.
No. of triangle
1
2
3
4
5
left-hand point
共..., ...兲
共..., ...兲
共..., ...兲
共..., ...兲
共..., ...兲
b Write down the coordinates of the left-hand points of triangles
6 and 7.
c The y-coordinate of the left-hand point of one of the triangles
is 10. What is the number of that triangle, and what is its
x-coordinate?
d The x-coordinate of the left-hand point of one of the triangles
is 66. Write down the y-coordinate of that point.
Combined Exercises
205
29 a Draw an axis with points C共3, 1兲 and D共⫺2, 2兲.
b Draw square ABCD, with points A and B below
the x-axis.
c Write down the coordinates of point A.
30 Draw an axis and colour all the grid points blue whose
y-coordinate is ⫺3, and whose x-coordinate is between
⫺1 and 4.
31 Copy the two tables below and fill in the blanks.
⫹
⫺12
⫺9
⫺7
⫺13
⫺10
0
5
⫺
4
⫺4
⫺23
⫺4
8
21
32 Calculate the following, and include the step in-between.
a ⫺5 ⫺ ⫺4 ⫹ 11
d ⫺75 ⫺ ⫺75 ⫹ 0
b 45 ⫺ ⫺45 ⫹ ⫺12
e 32 ⫺ ⫺32 ⫹ ⫺65
c ⫺21 ⫹ ⫺34 ⫹ ⫺21
f 0 ⫺ ⫺23 ⫹ ⫺34
33 Calculate.
a ⫺ 14 ⫹ ⫺ 45 ⫺ ⫺ 23
c ⫺ 38 ⫺ ⫺ 37 ⫹ ⫺ 14
b ⫺ 35 ⫺ ⫺ 35 ⫹ ⫺ 45
d ⫺ 37 ⫹ ⫺ 49 ⫺ ⫺ 23
34 With a height of 5,895 metres, Mount
Kilimanjaro is the tallest mountain in Africa.
The deepest point in the ocean is Vityaz Deep
in the Mariana Trench, which is 11,035 metres
deep.
What is the difference in height between the
peak of Mount Kilimanjaro and the bottom of
Vityaz Deep?
Write down the calculation relating to your
answer.
206
Combined Exercises
0
4 Diagrams
35 a What does the bar chart below tell you about France?
EUROPEAN ICE-CREAM CONSUMPTION
In liters per head of population
Sweden
12,2
Denmark
8,2
Great Britain
7,3
Ireland
7,3
Switzerland
5,9
The Netherlands
5,7
Belgium
5,7
Italy
5,7
Germany
5,6
France
4,9
Austria
4,2
Greece
3,3
Portugal
3,3
Spain
2,8
0
2
4
6
8
10
12
14
figure C.13
b Which Europeans consume the most ice-cream?
c Calculate how many litres of ice-cream the entire Dutch
population eats annually.
d Which Europeans eat approximately twice as much ice-cream
as the Spaniards?
e Give your comments on the following statements.
Combined Exercises
207
36 For some years, maths teacher Mr Kronenberg has been keeping
a record of the average weight of first-year pupils’ satchels.
The results are shown in the graph below.
Average weight of sarchels
weight in kg
11
10
9
8
O
'94
'95
'96
'97
'98
'99
'00
'01
'02
year
figure C.14
a What was the average weight of first-year pupils’ satchels in
school-year 96/97?
b What was the average weight in school-year 01/02?
c Which year showed the largest increase in weight? By how
many kg?
d Why has a tear line been used in the graph?
e According to Mr Kronenberg, the average satchel of a secondyear pupil is 3 kg lighter than that of a first-year pupil. Draw a
graph of the average weight of second-year pupils’ satchels.
37 The price-list on the right shows the tariffs of a
parking garage.
a Carlo parked his car for an hour and a quarter.
What was his parking fee?
b Marlous parked her car at 16.50 hrs. and
collected it again at 19.30 hrs.
How much did she have to pay?
c The garage manager has made a graph of
parking fees, so that he can quickly answer
customers’ queries.
Draw his graph, with the horizontal axis
extending to a parking period of 5 hours.
d Using the graph, pinpoint how much 3 hours
and 20 minutes parking would cost.
208
Combined Exercises
figure C.15
38 Martijn has malaria. This disease is common in
the tropics, and is spread by mosquitoes. The
fever pattern of malaria is shown in figure C.16.
a Is this a periodic graph? If so, how long is
each period?
b Martijn’s fever started on 14 May.
His birthday is on 21 May. Will he have a
temperature on his birthday? Explain your
answer.
MALARIA - TYPICAL FEVER PATTERN
The solid line shows the rise and fall of temperature
41˚
40˚
39˚
38˚
37˚
36˚
1
2
3
4
5
6
7
8
9
days of illness
39 Remco is a passenger in his mother’s car.
The car drives 300 metres in 12 seconds.
Draw up a relationship table that will show the
car’s speed in kilometres per hour.
figure C.16 Taken from an English medical book.
40 Garden centre FLOWER sells bags of flower bulbs.
Look at the table below.
No. of bags
1
2
3
4
5
6
7
8
price in €
3
6
9
12
14
16
18
20
a Draw a graph to represent this table.
b After how many bags does the price decrease?
D
50
50
20
50 45
41 Rondkerk municipality wishes to have as little traffic as possible
passing through the centre of town. This is why they have built
two roundabouts and introduced one-way traffic on many of the
streets. Look at the map on the right.
Copy the distance table below and fill in the blanks.
45 50
20
20
E
C
20
45
100
45
100
40
TO
DISTANCE
A
B
C
D
45
E
45
20
A
F
B
20
B
50 45
FROM
C
D
45 50
20
50
50
A
E
figure C.17 Distance table
figure C.18 Rondkerk town centre.
All distances are shown in metres.
Combined Exercises
209
42 There are two types of Dutch people, those who move house from time to time,
and those who never move. Amongst the type that move, one can distinguish
between those who move within the same municipality or province, and those
who move to another part of the country. This exercise concerns the last category.
For the sake of simplicity, we have divided the Netherlands into South, West,
North and East. Figure C.19 is an extract from the 1990 STATISTICAL YEAR-BOOK.
NORTH
2044
1381
4567
720
EAST
WEST
224
96
SOUTH
figure C.19 Balances of national migration between parts of the country in
1988. (Source CBS)
The arrow from West to South with the number 96 next to it,
means that 96 more people moved from the West to the South in
1988, than in the opposite direction.
a If a total of 3,560 people moved from the West to the South in
1988, how many moved from the South to the West?
b The graph in figure C.20 represents figure C.19.
Copy the graph and complete it.
c Which part of the Netherlands received the largest influx of
people due to moving house? Which part was depleted the
most?
210
Combined Exercises
N
720
W
E
S
4567
figure C.20 Incomplete graph.
5 Lines and angles
43 Figure C.21 shows a drawing of the Palace
Chapel in Aachen. Five lines have been marked
on the drawing.
a Line d is not horizontal. Explain why not.
b Which lines are horizontal?
c Which lines are vertical?
The palace of Charlemagne
The Palace Chapel in the German city of
Aachen was dedicated in 805 A.D., and is
the only part of Charlemagne’s palace still
standing. The chapel is still in its original
state.
The roof of this massive and sombre building
is supported by heavy square pillars. To
allow a view of the interior, part of the
building is shown in open section.
figure C.21
44 a Plot points A共⫺2, 0兲, B共6, 4兲, C共0, 4兲 and D共4, 0兲.
b
c
d
e
f
Draw line l through points A and B.
Draw line m through points C and D.
Draw line p through C, perpendicular to line l.
Draw line q through B, perpendicular to line m.
Draw line r through A, parallel with line m.
Lines l and m intersect at point S. Write down the coordinates of S.
Draw 䉭BCS. Measure the angles of this triangle and write them down.
45 Figure C.22 shows a drawing of a fire engine,
on a scale of 1 : 50.
, measure the angle
a In your
formed by the ladder and the horizontal line.
b Measure how long the ladder is in reality.
c The ladder can be extended to a length of
15 metres, and forms an angle of 55° with
the horizontal line. Make a drawing of this
situation on a scale of 1 : 150.
figure C.22
Combined Exercises
211
46 In quadrangle KLMN, ⬔K ⫽ 65°, ⬔L ⫽ 100°, KL ⫽ 6 cm,
KN ⫽ 3 cm, and LM ⫽ 4 cm.
a Draw quadrangle KLMN.
b Measure ⬔M and ⬔N.
47 The horse in figure C.23 has blinkers on, which
allow him a viewing angle of only 60°.
a Draw the horse’s viewing angle in your
.
b Which of the trees can the horse see?
48 A river runs right through the centre of a city.
Town Hall G and supermarket S are situated
on the same side of the river. Post Office P is
somewhere on the other side of the river.
figure C.23
Points G, S, and P form 䉭GSP.
⬔G of 䉭GSP ⫽ 50°, and ⬔S ⫽ 56°.
.
a Mark the position of the Post Office in your
b What is the distance in metres between the Post Office and the
Town Hall?
figure C.24 1 cm = 100 m.
49 How wide is the angle between the hands of a clock at:
a 11 o’ clock
b 7 o’ clock
c quarter to eight
212
Combined Exercises
d a quarter past eight
e twenty past two
f twenty-five past two
figure C.25
50 Hanneke has four 3.6 m long pieces of wire.
a With one of the pieces of wire, she is going to construct a wire model of
a cube, whose edges are to be as long as possible.
How long will each of the edges be?
b Using another piece of wire, Hanneke constructs a wire model of a
pyramid. The base is square, and all the edges are equally long, as well
as being as long as possible.
How long are the edges of this wire model?
c Hanneke is also going to construct a wire model of a prism, whose base
is to be an equilateral triangle. All its edges are to be equally long, and as
long as possible.
How long will the edges be?
d Hanneke constructs a cuboid from the last piece of wire. She makes two
of the faces square. The other sides are rectangles, whose longest sides
are twice the length of the shortest ones. Hanneke again uses up the
entire length of wire for this model.
How long are the edges of this cuboid? 共There are two possibilities.兲
51 a Draw cuboid ABCD EFGH, with AB ⫽ 6 cm, BC ⫽ 4 cm,
and AE ⫽ 5 cm.
b Write down whether the following pairs of lines are
intersecting, parallel or crossing.
AB and DH
EG and FH
FH and AC
AB and EF
BF and DH
BE and FH
Combined Exercises
213
Glossary
1 Space
Engels/English
Nederlands/Dutch
centre
middelpunt
circle
cirkel
cylinder
cilinder
diagonal
diagonaal
diameter
diameter
diameter 共ook!兲
middellijn
edges
ribben
faces
grensvlakken
flat shape
vlakke figuur
graph paper
roosterpapier
intersection
snijpunt
lines of vision
kijklijnen
net
uitslag
radius
straal
rectangle
rechthoek
right angle
rechte hoek
space figures
ruimtefiguren
square
vierkant
top view
bovenaanzicht
2 Numbers
214
Engels/English
Nederlands/Dutch
decimal number
decimaal getal
decimal place
decimaal 共achter de komma兲
denominator
noemer
Glossary
fraction
breuk
fractions with the same
denominator
gelijknamige breuken
integer
geheel getal
number line
getallenlijn
numeral
cijfer
numerator
teller
product
product
quotient
quotiënt
ratio
verhouding
ratio table
verhoudingstabel
round off
afronden
round up
afronden naar boven
round down
afronden naar beneden
sum
som
term
term
3 Locating points
Engels/English
Nederlands/Dutch
axis
assenstelsel
coordinates
coördinaten
grid point
roosterpunt
segment
lijnstuk
negative numbers
negatieve getallen
origin
oorsprong
positive numbers
positieve getallen
problem
probleem
x-coordinate
x-coördinaat
Glossary
215
x-axis
x-as
y-axis
y-as
y-coordinate
y-coördinaat
4 Diagrams
Engels/English
Nederlands/Dutch
bar chart
staafdiagram
diagram
graaf
directional diagram
gerichte graaf
flowing curve
vloeiende kromme
general graph
globale grafiek
link
verbinding
period
periode
periodic graph
periodieke grafiek
pictogram
beelddiagram
tear line
scheurlijn
5 Lines and angles
216
Engels/English
Nederlands/Dutch
acute angle
scherpe hoek
angle
hoek
base
grondvlak
crossing lines
kruisende lijnen
course 共heading, direction兲
koers
degrees
graden
flat angle
gestrekte hoek
intersecting lines
snijdende lijnen
Glossary
parallel lines
evenwijdige lijnen
parallel faces
evenwijdige vlakken
perpendicular
loodlijn
protractor
gradenboog
right angle
rechte hoek
sides
benen
top
bovenvlak
vertex
hoekpunt
perpendicular
loodrecht
wide angle
stompe hoek
wire model
draadmodel
7 IMA Mathematics and art
Engels/English
Nederlands/Dutch
motif
motief
patterned surface
vlakvulling
Glossary
217
Index
A
acute angle
angle
anamorphosis
applet
draught-board
turning
graphs
broken calculator
tile patterns
nets
Arabic numerals
arrow scale
axes
161
161
102
11
175
113
48
196
11
186
88
80
B
Descartes, René
diagonal
diagram
directional
diameter
directional diagram
divisor
draught-board 共applet兲
Dutch Railways 共NS兲 route
planner
81
19
127
132
21
132
69
11
135
E
edge
13
Escher, Maurits Cornelius 194
F
base
172
broken calculator 共applet兲 110
C
circle
computer program
coordinates
angles
‘This carries weight’
cone
constant
coordinates
coordinates 共program兲
course
crossing lines
cube
cuboid
cylinder
21
78
164
150
7
115
81
78
187
174
7
7
7, 21
factor
fall
flat angle
flat shape
fraction
34
115
161
12
42
G
general graph
graph
course of - from a table
periodic graph paper
graphs 共applet兲
grid
elongated curved slanted grid point
115
127
115
118
124
13
113
101
101
101
101
81
D
decimal number
decimals
decimal place
degrees
degrees 共program兲
denominator
218
Index
38
38
38
162, 168
164
42
H
hieroglyphs
68
I
impossible figure
integer
197
34
intersection
intersecting lines
22
174
L
lines
intersecting parallel crossing lines of vision
link
174
174
174
4
130
M
meteorologist
77
Möbius, August Ferdinand 192
- strip
192
motif
194
N
negative numbers
net
- of a cube
- of a pyramid
- of a cuboid
nets 共applet兲
75
10
10
10
17
11
O
origin
80
P
parallel
parallel lines
parallel faces
patterned surface
period
periodic graph
perpendicular
pictogram
positive numbers
prime number
prism
problem solving
154
155, 174
176
194
124
124
153
110
75
69
7
104
product
protractor
protractor triangle
pyramid
34
164
19
7
Q
quadrant
quotient
103
34
R
radius
ratios
ratio table
rectangle
reduce
right angle
Roman numerals
rounding off
21
56
57
19
43
19, 161
68
41, 52
S
same denominators
44
scale
38
segment 共of line兲
82
separate the integers from
the fraction
43
side
161
space shapes
7
sphere
7
square
12
sum
34
super cube
30
surveyor
166
T
tear line
‘This carries weight’
共program兲
tile patterns 共applet兲
top
top view
turning 共applet兲
118
150
196
172
4
175
Index
219
V
X
vertex
vertical
161
153
x-axis
x-coordinate
161
y-axis
y-coordinate
W
Y
wide angle
220
80
81
Index
80
81
Illustrations
Fotoresearch: Bureau voor beeldresearch ELF / Elvire Berens, Geldermalsen
Illustrations acquired by: Haasart, Wim de Haas, Rhenen
Technical drawings: Buro van Dulmen, Veldhoven
Illustrations
ABC Press/J.M. Loubat, Amsterdam: p. 75
AKG, Berlin: p. 196
The Ancient Art & Architecture Collection, Pinner 共GB兲: p. 72
ANP-Foto, Rijswijk 共ZH兲: p. 113
Bibliotheek Technische Universiteit Eindhoven: fig 4.57
Corel Cooperation, Salinas 共VS兲: p. 210
Hilbert Bolland, Breda: pp. 45, 83, 97, 139, fig 5.14, 5.15
Marco van Bergen, Baarn: p. 174, fig 1.68, 1.69, 2.9, 2.22, 3.26, 4.20, 5.33, IMA.2, IMA.3, IMA.4,
IMA.5
Fotodienst Vliegbasis Leeuwarden: p. 81
Fotostock, Amsterdam: pp. 77, 159, fig 1.12
Haasart, Wim de Haas, Rhenen: pp. 25, 53, 117, 143, 166, fig 21
Hollandse Hoogte, Amsterdam: fig 5.1
H. Jobse, Culemborg: fig 2.37
Dirk Kreijkamp, Den Bosch: fig 1.39
Gerrit de Jong, Middelburg: p. 55, fig 1.12, 1.37, 1.53, 1.67, 2.8, 2.16, 2.17, 2.21, 2.33, 5.29, 5.50
Marcel Jürriens, Boxtel: p. 161, fig 1.3, 1.54, 1.61, 2.32, 3.1, 4.71, 5.14, 5.20, 5.35, 5.72
Klutworks, Dik Klut, Den Haag: To the pupil, fig 2.1, Whizz-kids: fig 3.6, 3.40, pp. 14, 21, 38, 40, 42, 44,
47, 49, 52, 54, 55, 56, 62, 63, 64, 65, 66, 67, 80, 84, 86, 89, 92, 103, 107, 108, 120, 121, 122, 159, 166,
169, 170, 172, 175, 206
Kyodo, Tokio: p. 75
Ligthart Fotografie, Amsterdam: pp. 37, 155, 173
Marjolein Luiken, Amsterdam: fig 4.25, 4.85
Bas de Meijer, Zevenaar: p. 70
Joop Mommers, Barendrecht: pp. 39, 42, 68, 82, 168, 184, fig 1.2, 1.5, 1.27, 1.48, 1.62, 2.6, 2.9, 2.16,
2.27, 3.16, 3.20, 4.24, 4.36, 4.55, 5.12, 5.15li-bo, 5.15re-on, 5.16, 5.17, 5.24, 5.74, 5.75, 5.81, C.18, C.26
P.P. de Nooyer/Foto Natura, Wormerveer: p. 94
Octopus Publishing Group Ltd., London: fig C.24
Picture Box, Wormerveer: pp. 116, 174
Ton Poortvliet, Dordrecht: fig 1.9, C.2
Jan Rijsterborgh, Haarlem: p. 171, fig 5.31
Pim Rusch Fotografie, Leiden: p. 170, fig 1.9, 1.28, 1.29, 5.38
Roeland van Santbrink, Bussum: p. 163
Vandystadt/Omnipress. Foto: J.M. Loubat, Den Haag: p. 94
Ben Verhagen, Schijndel: pp. 22, 40, 62, 80, 81, 91, 95, 96, 126, 164, 167, 211, fig 1.10, 1.23, 2.3, 2.28,
5.62, 5.71, 6.2, 6.3
Zanzara, Marcel Braat, Odiliapeel: fig 2.4, 3.21
Möbius Strip II by M.C. Escher. 共c兲2002 Cordon Art, Baarn, Holland. All rights reserved: p. 196
Regular patterned surface E96 by M.C. Escher. 共c兲 2002 Cordon Art, Baarn, Holland. All rights reserved:
p. 198
Regular patterned surface E22 by M.C. Escher. 共c兲 2002 Cordon Art, Baarn, Holland. All rights reserved:
p. 198
Belvedere by M.C. Escher. 共c兲 2002 Cordon Art, Baarn, Holland. All rights reserved: p. 201
Illustrations
221
214678

numbers and space

Transcription

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