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Sample pages from the new edition of Beta Mathematics,
showing:
9
New investigations (Prime Boeings)
80
Ratio – investigating RGB
119
Use of colour to enhance algebraic patterns diagrams
129
Use of colour to make common factors obvious
171
BMI investigation
175
New applications throughout – plenty of work for students to do in class
211
New applications throughout – plenty of work for students to do in class
300
Subtle use of colour
346
SOH-CAH-TOA, by popular request!
416
Evaluating Statistical Reports – focus of new curriculum
458
The Statistical Enquiry Cycle (PPDAC)
461
Multivariate data
1 Number properties and operations
PUZZLE
Prime Boeings
© Photographer: Ivan Cholakov | Agency: Dreamstime.com
The Boeing company, based in Seattle, WA, United States, is the world’s largest aeroplane
manufacturer. Their passenger aircraft have model names such as 7x7.
First flight
1954
1998
1963
1967
1969
1982
1981
1994
2007
© Photographer: Ramon Berk | Agency: Dreamstime.com
Model
707
717
727
737
747
757
767
777
787
© Photographer: Ramon Berk | Agency: Dreamstime.com
Use appropriate technology to answer these questions:
1
2
01SD Beta.indd 9
Write down all the prime Boeing model numbers.
Write down each of the composite (non-prime) Boeing model numbers
as a product of prime factors.
20/1/08 1:02:34 PM
80
Number and algebra
12 Henry and Rose are each sanding a particle-board floor.
Henry sands an area of 12 m2 in 15 minutes. Rose sands
an area of 18 m2 in 20 minutes.
a Express the rate in m2/h at which each person
works.
b Who can sand at the faster rate?
13 Grass seed is to be spread on
a farm at a rate of 240 kg/ha.
How many hectares can be
sowed by 2 tonnes of grass
seed? (Note: 1 tonne = 100 kg.)
Investigation
Colour-mixing (RGB)
B
Ratios are used to create and define different colours.
Colours can be specified by how many parts
out of 255 are used for each of red, green and blue.
The letters RGB are used together with numbers
in a three-way ratio to show the exact proportions.
For example, the primary colour yellow, at its most
light and intense, is 255 : 255 : 0.
It would be darkened by lowering the numbers to
205 : 205 : 0, but the ratio of red to green is still
the same, and there is no blue at all in the mixture.
R
G
A colour that has a mixture of all three of red, green and blue, for example, ‘sienna’, has an RGB
ratio of 255 : 135 : 71.
On a TV set, to show white, all three types of pixel (red, green and blue) are turned on. The RGB
code for white is 255 255 255.
Visit www.mathematics.co.nz for links to websites that give examples
of RGB colour-mixing.
• http://www.pitt.edu/~nisg/cis/web/cgi/rgb.html
• http://web.njit.edu/~kevin/rgb.txt.html
• http://www.lon-capa.org/~mmp/applist/RGBColor/c.htm
• http://mc2.cchem.berkeley.edu/Java/RGB/example1.html
• http://www.etntalk.com/colorpicker/flash/colorpicker.swf
05Beta.indd 80
20/1/08 7:34:25 PM
119
8 Simplifying algebraic expressions
4 a
b
c
Copy this pattern. Continue it for two
more steps.
How many cubes are added on at each
step?
Write down a rule that describes how
many cubes there are at each step, in the form c = ____.
Step (n)
Number of
cubes (c)
5 a
b
c
Copy and continue this
pattern for two more
steps.
How many matchsticks
are added on at each
step?
Write down a rule that
describes how many
matchsticks there are at
each step, in the form
m = ____.
1
2
1
3
Step (n)
1
Number of
matchsticks (m) 6
2
3
3
9
6 The drawing shows several flags
strung up along a line of lamp-posts.
There are exactly four flags between
each lamp-post.
a
b
c
d
e
Copy and complete this table.
Timotei thinks the rule is f = (l − 1)2. Give a
reason why this rule is wrong.
Write down the correct rule linking l and f,
in the form f = ______.
Use the rule to work out the number of
flags if there are 29 lamp-posts.
A line of lamp-posts has 204 flags strung
up. How many lampposts are there?
7 This diagram shows an arrow
that is gradually growing.
a Copy and complete this table.
n
Number of
squares (s)
b
08Beta.indd 119
1
1
2
4
3
7
4
n=1
5
Number of
lamp-posts (l)
1
2
3
4
5
6
n =2
n=3
Number of
flags (f)
16
n=4
n=5
6
How many squares are added on at each step?
c
Write down a rule that describes how
many squares there are at each step, in
the form s = ______.
20/1/08 7:58:45 PM
129
9 Expanding and factorising
The expression to be factorised can have some terms made up of letters, and other terms consisting of
numbers only.
Example
Factorise 6x + 21.
6x is made up of factors 3, 2 and x.
21 is made up of factors 3 and 7.
3 is a common factor. Inside the brackets we write
the terms left behind after the 3 has been taken out.
6x + 21 = 3(2x + 7)
Example
Factorise 24x − 32.
24x − 32 = 8(3x − 4)
TIP
Answer
Factorise 8x − 4.
EXERCISE
Always
take out
the highest
possible
common
factor.
Answer
Example
TIP
Answer
The common factor is 4. Note: 4 = 4 × 1,
so we leave a 1 inside the brackets.
8x − 4 = 4(2x − 1)
Sometimes the term
left behind inside the
brackets will just be
the number 1.
9.05
Factorise these expressions.
1 3x + 6
8
2 4x + 8
9
3 6x + 8
10
4 8x + 12
11
5 12x − 8
12
6 3x + 30
13
7 4x + 6
14
21x + 14
4x + 8
6x + 9
5x − 15
24x − 16
5x + 5
7x − 7
15
16
17
18
19
20
21
4x + 2
15x − 21
14x + 35
16x − 4
15x − 5y
46x + 23
60x − 90
22
23
24
25
26
27
28
30x − 5
45x + 30
6x − 9y + 12z
24p − 18q + 30r
3a + 6b + 18
4x + 4y − 4
40x + 8y + 4
Letters as common factors
Example
Factorise ab + ac + ad + 2a
Answer
The common factor is a. It is written in front of the brackets.
ab + ac + ad + 2a = a(b + c + d + 2)
You must always take
out as many common
factors as possible. For
Answer
the example here,
abc + abcd + 12ab = ab(c + cd + 12)
a(bc + bcd + 12b) would
(The common factor is ab)
be wrong. This is because
TIP there is still the letter b
that can be factorised out
of the brackets.
Harder examples have 2 letters or more as common factors
Example
09Beta.indd 129
Factorise abc + abcd + 12ab
20/1/08 7:53:58 PM
12 Two dimensional graphs
171
BMI-for-age percentiles: girls, 2–20 years
32
95th
30
28
85th
26
22
50th
20
18
Percentile
BMI (kg/m2)
24
5th
16
14
12
2 3
4
5
6
7 8
9 10 11 12 13 14 15 16 17 18 19 20
Age (years)
Body-fatness description
Percentile range
Obese
Above the 95th percentile
Overweight
Between the 85th and 95th percentile
Healthy weight
Between the 5th and 85th percentile
Underweight
Below the 5th percentile
Note: the data in the table is based on data collected from 33 445 children in the US from 1963 to 2002. The BMI does
not distinguish between fat, muscle and bone. The BMI ranges should be treated cautiously for some groups, including
Asians (who often have small bones) and Pacific Islanders (who often have large bones). Sources of data: National
Health and Nutrition Examination Survey.
1 What colour represents the band for
children with a healthy BMI?
2 a
b
12Beta.indd 171
A 14-year-old girl has a BMI of 19.
What body-fatness description fits
the girl?
An 11-year-old boy has a BMI of 22.
What body-fatness description fits
the boy?
3 A 13 1 -year-old boy has a height of
2
1.48 m and a weight of 49 kg. Calculate
his BMI and use the result to describe the
boy’s body-fatness.
4 Between what age ranges do children
appear to have the lowest BMI?
20/1/08 7:45:14 PM
175
12 Two dimensional graphs
The first distance–time graph (A) shows the journey for a student who walks slowly, realises she
has forgotten her lunch, runs home, makes the lunch, and runs to school.
Describe a possible journey for each of graphs labelled (B) to (E).
Distance from Pipiriki (km)
5 Two groups of kayakers are travelling along the
Whanganui River, camping at the same places
54
each night. Yesterday they travelled a distance of
54 km. The graph shows the distance from Pipiriki
48
at different times during the day. One group has
42
red kayaks, the other has orange kayaks. Each
36
group stopped at the same time for lunch. One of
the groups was able to drift downstream without
30
paddling for some of the journey.
24
a Which group left Pipiriki first?
18
b Which group reached the night-time
campground first?
12
c When did each group stop for lunch, and how
6
far apart were the two lunch stops?
d When on the journey did the two groups meet?
10 11 12 1 2 3 4 5 6
am
pm
e What was the greatest separation (in
Time of day
kilometres) between the groups?
f The orange group left behind some gear at the
lunch stop and had to return to collect it. How long did it take them to return to the lunch
spot after they noticed the gear was missing?
g How far did the orange group travel altogether?
h Which group was able to drift downstream without paddling?
i Use the information from the graph and write down a calculation to show that the river
flows at 2.4 km/h.
j Give a reason why the orange group were travelling most slowly between 2 pm and 3 pm.
12Beta.indd 175
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000
0900
0800
0700
6 A small airline operates two
Dunedin
Keruru
planes, named Keruru and Kea,
Kea
between Auckland, Wellington,
Christchurch
Christchurch and Dunedin.
a There is one flight a
Wellington
day from Dunedin to
Christchurch. When does
Auckland
it leave Dunedin?
b How long does it take
to fly from Dunedin to
Wellington?
Time of day
c How many direct flights
are there from Auckland to
Christchurch each day?
d Which route is faster to fly from Dunedin to Auckland – via Wellington or via Christchurch?
e Where are the planes kept overnight?
f Are the planes ever timetabled to be at the same airport at the same time? Explain.
20/1/08 7:45:23 PM
211
14 The metric system, scales and tables
8 On some flights you can view an Airshow.
This displays the progress of the flight,
including the route and information about
the time remaining until arrival at the
destination.
Time to destination
Estimated arrival time
Local time at destination
10 Flight NZ791 takes off from Christchurch at
0610 (local time) and lands at Melbourne at
0800 (local time). The time in Christchurch is
2 hours ahead of the time in Melbourne.
0.48
0730
0642
In each case below calculate the values for a,
b and c.
a
Time to destination
Estimated arrival time
1130
0745
Local time at destination
Melbourne
Christchurch
Time to destination
Estimated arrival time
Local time at destination
6.38
b
0327
a
b
Time to destination
Estimated arrival time
Local time at destination
3.17
2104
c
9 Flight ZQ455 is timetabled to leave
Auckland at 1355 and arrive at Wellington
at 1455. It continues from Wellington at 1610
and arrives at Dunedin at 1705.
a For how long is the plane on the ground
at Wellington?
b What is the total flying time for the
journey from Auckland to Dunedin?
14Beta.indd 211
What is the duration of the flight from
Christchurch to Melbourne?
The return eastbound flight NZ792 is
half an hour shorter than the westbound
flight because of favourable winds.
If it takes off from Melbourne at 0900
(local time), what would be the time in
Christchurch when it lands?
11 A video recorder can rewind 72 times faster
than it plays. An E-240 video tape takes
4 hours to play from beginning to end. How
long, in minutes and seconds, does it take to
completely rewind?
12 Julie programmes her DVD-recorder to start
recording at 2055 and finish at 2210. How
long will the recording be?
15/2/08 12:38:50 AM
300
Measurement and Geometry
21
Three dimensions
STARTER
It has always been a challenge to show three-dimensional (3-D) objects on a flat twodimensional (2-D) surface, like a movie screen or the page of this book.
Here are some ‘phantasmagorical’ shapes.
P
The Klein bottle
Q
1
2
3
4
The impossible triangle
The level staircase
How many surfaces does a Klein bottle have?
Does the Klein bottle have an inside or an outside?
In the impossible triangle is the cube marked P above or below the cube marked Q?
Are the top and the bottom of the staircase at the same level?
Interpretation of three-dimensional shapes
The simplest three-dimensional shape to visualise is a cuboid. These are two ways of representing a
cuboid with measurements 5 units by 3 units by 2 units.
Isometric drawing
21Beta.indd 300
Oblique drawing
15/2/08 12:52:36 AM
346
Measurement and Geometry
24
Trigonometry 2 - calculating any side length
To learn trigonometry today will have an effect tomorrow.
2 learn + 2 day = 4 tomorrow
We will start with a summary of trigonometry so far.
o = sin(A) a = cos(A)
h
h
)
(h
en
ot
p
Hy
e
us
Opposite(o)
A
If we are not working with the length of the hypotenuse,
we use the tan ratio:
o = tan(A)
a
Adjacent(a)
TIP
Here is some advice on remembering these three formulae.
sin( A) = o
cos( A) = a
tan( A) = o
a
h h The key letters are:
SOH
CAH
TOA
Use some ‘triangles of facts’ to show how the formulae are set out:
sin
S
24Beta.indd 346
o
=
h
O H
cos
a
=
h
C A H
tan
T
o
=
a
O A
15/2/08 12:58:25 AM
416
Statistics
a
b
c
d
e
Use the graph to estimate how much was spent on health per person in New Zealand in 2007.
Which countries spent more than twice the amount per person on health than New Zealand
did in 2007?
Which country had the closest GDP per person to New Zealand?
Which countries spent about the same on health expenditure per person as New Zealand?
Explain what the graph shows about the relationship between GDP and health
expenditure per person for these countries.
Statistical reports
A useful skill in today’s world is to be able to assess a statistical report and decide whether the
conclusions are justified, given the data. When ‘thinking statistically’ you should take these factors
into consideration:
• Where and how was the data collected?
• Did all the people of interest have the same chance of being studied, or were certain groups more
likely to be chosen, and if so, does that influence the result?
• Can the conclusions be supported from the data?
New Zealand Curriculum note:
students should be able to critically
evaluate data-based arguments in
media and other sources.
STARTER
Here is a
controversial
example from
the front page of
The New Zealand
Herald, which shows
a wrong conclusion
about risk factors
for DVT (deep vein
thrombosis) or
blood-clotting in
the veins. See if you
can spot the faulty
reasoning.
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458
Statistics
32
The statistical enquiry cycle
Introducing PPDAC
IS
A N A LYS
A
D
Understand and define a statistical problem, usually in the form
of a question. How would you answer the question?
• A summary question: ‘I wonder … what proportion of
students in my class are given pocket money’.
• A relationship question: ‘I wonder if … there is a relationship
between the weight of a student’s bag and their year level’.
• A comparison question: ‘I wonder if … the boys at my school
send more text messages than the girls at my school’.
N C L U SI O N
LEM
1 The problem
CO
OB
PR
In this chapter your job is to act as a ‘data detective’ - in other
words, to carry out a statistical enquiry or investigation. There are
five steps which can be summarised by the letters PPDAC.
TA
PL
A
N
2 The plan
What are you going to measure and how?
• Design a study.
• How will you collect and record your data - what
measurements do you need to take, where from, and when?
3 The data
Now you have your data you need to organise it, and you may
need to check it.
• Decide whether to sort the data.
• Does the data need ‘cleaning’? Are there measurement or
recording errors?
• Will you use all of the data?
4 The analysis
Use the data to help answer the question posed at the beginning.
• You will need to summarise your data in some way. Construct tables and graphs to present the
data in an easy-to-understand way.
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461
32 The statistical enquiry cycle
Student
number Year Sex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
9
10
10
10
10
9
9
10
10
9
9
10
10
9
9
9
10
9
10
10
10
10
9
10
10
9
9
10
9
10
10
10
9
9
10
9
9
9
9
9
F
M
F
F
F
F
M
M
M
F
F
M
F
F
M
M
F
M
F
F
M
F
F
M
M
M
M
F
M
M
F
M
M
M
M
F
M
Yes
F
M
Ethnic
group
NZ Pakeha
Asian
Asian
NZ Pakeha
Other
NZ Pakeha
NZ Pakeha
NZ Pakeha
Asian
NZ Pakeha
NZ Pakeha
Maori
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
Asian
NZ Pakeha
NZ Pakeha
Other
Maori
Asian
NZ Pakeha
Other
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
Asian
NZ Pakeha
Asian
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
NZ Pakeha
Maori
Pocket Income from
Weight Height Wrist Neck Hand-span Reaction money part-time job Internet
(kg)
(cm) (mm) (cm)
(mm)
time (s)
($)
($)
at home?
49
44
41
46
48
57
66
68
52
41
47
66
119
55
67
60
39
44
61
47
46
42
42
79
55
45
43
61
51
4200
46
53
45
49
52
44
55
49
60
51
168
154
156
162
173
154
1698
170
168
153
165
162
171
160
181
169
149
176
172
150
156
156
153
169
154
150
152
176
177
158
150
166
162
179
164
179
1.79
161
177
161
178
171
164
171
193
175
171
190
169
160
165
181
177
168
201
187
162
176
172
165
172
170
163
175
167
152
160
196
188
183
166
167
16
202
189
192
181
180
181
164
36
38
37
36
37
35
38
40
38
36
36
40
37
37
41
40
35
37
39
37
37
37
36
41
39
36
34
41
37
38
36
38
37
35
38
37
38
35
39
39
195
178
178
192
212
178
203
207
198
171
193
193
210
187
236
202
153
215
211
161
184
180
171
204
178
158
163
216
218
187
161
193
152
227
193
221
232
191
218
192
0.8
0.7
0.5
0.7
0.5
0.4
0.9
0.5
0.9
0.8
0.5
0.7
0.6
0.4
0.6
0.5
0.9
0.8
3
0.5
0.4
0.4
0.7
0.5
0.5
0.5
0.6
0.4
0.4
0.9
0.5
0.4
0.8
0.9
0.4
0.4
0.9
0.8
0.5
0.4
40
0
10
50
50
60
60
0
20
25
0
50
10
60
0
15
0
30
0
40
20
10
0
50
40
10
50
100
30
30
20
0
60
80
35
40
10
50
25
25
61
46
32
0
35
38
39
48
35
47
38
22
51
0
26
66
26
40
52
0
54
32
31
80
32
22
0
0
42
58
29
71
17
0
33
36
24
47
43
68
Broadband
Broadband
Broadband
Broadband
Broadband
Broadband
Broadband
No
Dial-up
Broadband
No
Broadband
Broadband
Broadband
Dial-up
Broadband
Broadband
Broadband
Dial-up
No
Dial-up
Broadband
Dial-up
Broadband
Broadband
Dial-up
Broadband
Broadband
Broadband
Dial-up
Broadband
Broadband
No
Broadband
Firefox
Broadband
Broadband
Broadband
Dial-up
Broadband
The figures for pocket money and income from a part-time job are weekly estimates. The wrist and neck measurements are
for the circumference. The reaction time is measured electronically by timing how long it takes a student to press a brake
simulator to the floor in response to a flashing light, and it times out after 3 seconds.
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