AS Level Use of Mathematics
IB Mathematical Studies
Transition activities
There are two pieces of work (Solar Eclipse and Linear Functions) to be
completed and brought with you to your first Maths lesson at the College.
SOLAR ECLIPSE
Data Sheets 1 and 2 below give information about total eclipses which have
occurred in the 20th century, Data Sheet 3 gives information about total
eclipses which are predicted for the first 35 years of the 21st century, Data
Sheet 4 gives more detailed information about the last total eclipse to
occur in the UK and Data Sheet 5 gives general information about types of
eclipse.
You should study the data sheets given and then answer the following
questions, giving calcualtions to support your answers:





Are total eclipses becoming more frequent?
What is the ‘average’ duration of a total eclipse?
Was the 1999 total eclipse ‘typical’?
If a histogram is drawn of the maximum path widths, is it symmetrical?
Do those eclipses which last for the longest times also tend to have the
widest path widths?
Data Sheet 1
Key * denotes a total annular eclipse
Total Solar Eclipses (1900 - 1949)
Date
Year Month Day
1900
1901
1903
1904
1905
1907
1908
1908
1909
1910
1911
1912
1912
1914
1916
1918
1919
1921
1922
1923
1925
1926
1927
1928
1929
1930
1930
1932
1934
1936
1937
1938
1939
1940
1941
1943
1944
1945
1947
1948
Mar
May
Sep
Sep
Aug
Jan
Jan
Dec
Jun
May
Apr
Apr
Oct
Aug
Feb
Jun
May
Oct
Sep
Sep
Jan
Jan
Jun
May
May
Apr
Oct
Aug
Feb
Jun
Jun
May
Oct
Oct
Sep
Feb
Jan
Jul
May
Nov
28
18
21
9
30
14
3
23
17
9
28
17
10
21
3
8
29
1
21
10
24
14
29
19
9
28
21
31
14
19
8
29
12
1
21
4
25
9
20
1
Maximum
Maximum
Duration Path Width
(min:sec)
(miles)
02:10
58
06:28
149
02:11
157
06:19
146
03:46
123
02:24
119
04:13
93
00:11
6
00:23
32
04:15
04:57
120
00:01
1
01:55
54
02:14
113
02:36
69
02:22
70
06:50
153
01:52
189
05:58
142
03:36
106
02:32
130
04:10
92
00:50
48
05:06
00:01
01:55
01:44
02:52
02:31
07:04
04:04
01:32
05:35
03:21
02:39
04:08
01:15
05:13
01:55
122
1
54
104
79
83
156
276
137
91
146
90
57
124
53
Area
Mexico, USA, Spain, Africa
Indian Ocean, Sumatra, Borneo, New Guinea
Indian Ocean, Antarctica
Pacific Ocean
Canada, Spain, Africa, Arabia
Russia, China, Mongolia
Pacific Ocean
S. America, Atlantic Ocean, Indian Ocean
*Greenland, Russia
Antarctica
Pacific Ocean
*Atlantic Ocean, Europe, Russia
Colombia, Brazil, Atlantic Ocean
Greenland, Europe, Middle East
Colombia, Venezuela, Atlantic Ocean
Pacific Ocean, USA
S. America, Atlantic Ocean, Africa
Antarctica
Indian Ocean, Australia
Pacific Ocean, California, Mexico
Great Lakes, USA, Atlantic Ocean
Africa, Indian Ocean, Borneo
England, Scandinavia, Arctic Ocean, Siberia
(Umbra hardly touched Antarctica)
Indian Ocean, Malaya, Philippines
*Pacific Ocean, USA, Canada
Pacific Ocean
Arctic Ocean, Canada, New England
Borneo, Pacific Ocean
Greece, Turkey, Soviet Union, Japan
Pacific Ocean, Peru
Atlantic Ocean
Antarctica
Colombia, Brazil, Atlantic Ocean, S. Africa
Soviet Union, China, Pacific Ocean
Japan, Pacific Ocean, Alaska
Peru, Brazil, W. Africa
USA, Canada, Greenland, Scandinavia, Soviet Union
S. America, Atlantic Ocean, Africa
Africa, Indian Ocean
Data Sheet 2
Total Solar Eclipses (1950 - 1999)
Date
Year Month Day
1950
1952
1954
1955
1956
1957
1958
1959
1961
1962
1963
1965
1966
1967
1968
1970
1972
1973
1974
1976
1977
1979
1980
1981
1983
1984
1985
1986
1987
1988
1990
1991
1992
1994
1995
1997
1998
1999
Sep
Feb
Jun
Jun
Jun
Oct
Oct
Oct
Feb
Feb
Jul
May
Nov
Nov
Sep
Mar
Jul
Jun
Jun
Oct
Oct
Feb
Feb
Jul
Jun
Nov
Nov
Oct
Mar
Mar
Jul
Jul
Jun
Nov
Oct
Mar
Feb
Aug
12
25
30
20
8
23
12
2
15
5
20
30
12
2
22
7
10
30
20
23
12
26
16
31
11
22
12
3
29
18
22
11
30
3
24
9
26
11
Key * denotes a total annular eclipse
Maximum
Maximum
Duration Path Width
(min:sec)
(miles)
01:13
90
03:09
89
02:35
96
07:07
159
04:44
269
05:10
03:01
02:45
04:08
01:39
05:15
01:57
131
76
164
92
63
124
53
00:39
03:27
02:35
07:03
05:08
04:46
02:37
02:49
04:08
02:02
05:10
01:59
01:58
00:01
00:07
03:46
02:32
06:53
05:20
04:23
02:09
02:50
04:08
02:22
68
99
111
160
216
125
63
195
93
68
125
53
430
1
3
109
130
160
182
117
48
221
94
69
Area
Arctic Ocean, Siberia, Pacific Ocean
Africa, Middle East, Soviet Union
USA, Canada, Iceland, Europe, Middle East
Indian Ocean, Southeast Asia, Philippines, Pacific Ocean
Pacific Ocean
(Umbra hardly touched Antarctica)
Pacific Ocean, Chile, Argentina
New England, Atlantic Ocean, Africa
Europe, Soviet Union
Borneo, New Guinea, Pacific Ocean
Pacific Ocean, Alaska, Canada, Maine
New Zealand, Pacific Ocean
Pacific Ocean, S. America, Atlantic Ocean
(Umbra hardly touched Antarctica)
Soviet Union, China
Pacific Ocean, Mexico, USA, Canada
Siberia, Alaska, Canada
Atlantic Ocean, Africa, Indian Ocean
Indian Ocean, Australia
Africa, Indian Ocean, Australia
Pacific Ocean, Colombia, Venezuela
USA, Canada, Greenland
Africa, Indian Ocean, India, Burma, China
Soviet Union, Pacific Ocean
Indian Ocean, Indonesia, New Guinea
New Guinea, Pacific Ocean
Antarctica
*Atlantic Ocean
*Atlantic Ocean, Africa
Sumatra, Borneo, Philippines, Pacific Ocean
Finland, Soviet Union, Aleutian Islands
Hawaii, Mexico, Central America, Colombia, Brazil
Atlantic Ocean
Peru, Bolivia, Paraguay, Brazil
Iran, India, Southeast Asia
Mongolia, Siberia
Galapagos Islands, S America, Caribbean
Europe, Middle East, India
Data Sheet 3
Predicted for the Future
Key * denotes a total annular eclipse
Total Solar Eclipses (2000 - 2035)
Date
Year Month Day
2001
2002
2003
2005
2006
2008
2009
2010
2012
2013
2015
2016
2017
2019
2020
2021
2023
2024
2026
2027
2028
2030
2031
2033
2034
2035
Jun
Dec
Nov
Apr
Mar
Aug
Jul
Jul
Nov
Nov
Mar
Mar
Aug
Jul
Dec
Dec
Apr
Apr
Aug
Aug
Jul
Nov
Nov
Mar
Mar
Sep
21
4
23
8
29
1
22
11
13
3
20
9
17
2
14
4
20
8
12
2
22
25
14
30
20
2
Maximum
Maximum
Duration Path Width
(min:sec)
(miles)
04:56
124
02:03
54
01:57
308
04:31
100
04:06
114
02:27
147
06:38
160
05:20
160
04:02
111
01:39
35
02:46
287
04:09
96
02:40
71
04:32
124
02:09
56
01:54
260
01:16
30
04:28
122
02:18
182
06:22
160
05:09
143
03:43
105
01:08
24
02:37
483
04:09
99
02:54
72
Area
Atlantic Ocean, Africa
Africa, Indian Ocean, Australia
Antarctica
*Pacific Ocean
Africa, Turkey, Soviet Union
Greenland, Soviet Union, China
India, China, Pacific Ocean
Pacific Ocean, S. America
Australia, Pacific Ocean
*Atlantic Ocean, Africa
Atlantic Ocean, Norwegian Sea, Svalbard
Indonesia, Pacific Ocean
USA
Pacific Ocean, Chile, Argentina
Chile, Argentina
Antarctica
*Indonesia
Mexico, USA, Canada
Greenland, Iceland, Spain
Gibraltar, Africa, Saudi Arabia
Indian Ocean, Australia, New Zealand
S. Africa, Indian Ocean, Australia
*Pacific Ocean
Alaska, Arctic Ocean
Africa, Middle East
China, Korea, Japan, Pacific Ocean
Data Sheet 4
11th August 1999
Did you see the eclipse? The map shows approximately how much of the sun was
eclipsed by the moon in regions of the U.K.
70%
75%
80%
85%
90%
95%
Total Eclipse
in this region
100%
The eclipse occurred in Europe and Asia, with a total eclipse in the following countries:
England, France, Germany, Switzerland, Austria, Hungary, Romania, Turkey, Iraq, Iran,
Pakistan and India.
Data Sheet 5
What happens during a solar eclipse
The umbra is the
region in total
shadow behind
the moon.
The shadow of the moon falls on the surface of the earth as shown in the sketch. This
shadow moves at a speed of more than 1000 mph from west to east across the surface of
the earth.
Types of Solar Eclipse
Total
The sun is about 400 times as big as the moon, but because it is about 400 times as far
away, they appear to be roughly the same size. In a total eclipse the moon appears to
move across the sun until it completely covers it. This happens at places crossed by the
shadow where the umbra meets the earth. The whole process can take a few hours, but
the period of totality lasts only a few minutes. In a wide area around the shadow the sun
is only partly eclipsed by the sun.
Partial
The orbit of the moon is at an angle to the orbit of the sun. This means that sometimes the
umbra passes the earth without touching it, but is near enough to give a partial eclipse in
some places. These eclipses are not included on the data sheets.
Annular
The shape of the moon’s orbit means that the distance between the moon and earth varies.
If the umbra comes to a point before it reaches the earth, the eclipse is annular. This
means that the moon is too small to completely eclipse the sun. When this happens the
part of the sun surrounding the earth looks like a very bright doughnut.
Example
A car travels at a steady speed of 50mph along a motorway.
The table shows the distances it would go in 1 hour, 2 hours, etc.
Time t hours
Distance d miles
0
0
1
50
2
100
3
150
4
200
Distance/Time Graph
distance (miles)
200
150
100
50
0
0
1
2
3
4
tim e (hours)
The distance can be found from the time by multiplying by 50. The points are on a
straight line.
The gradient of the line is 50. The relationship between t and d is linear.
The distance, d, is directly proportional to the time t. [written d  t ]
Task 1
The mass of a cubic centimetre of iron is 8 grams
(a) Copy and complete the table.
Volume cm3 1
2
Mass g
8
16
3
24
4
?
(b) Draw a graph to show the relationship. Check that the points are on a straight line.
(c) What is the gradient of the line ?
(d) When the volume is multiplied by 2, what happens to the mass ?
(e) What is the relationship between the volume and the mass ? Write the relationship in
symbols.
Task 2
1) Investigate the following relationships and determine which are linear.
Comment on key features of any graphs drawn and where possible write the
relationship in symbols.
(a) The table below shows the frequencies of the notes obtained from a stretched
sting under various tensions.
Tension N
Frequency Hertz
16
200
36
300
49
350
64
400
(b) A small ball is thrown upwards with various initial speeds. The height it goes
is recorded below.
Speed ms-1
Height m
(c)
8
3.2
18
16.2
22
24.2
27
36.45
The current through a resistor when connected to various cells was:
Voltage V Volts
Current I Amperes
1.5
0.3
3
0.6
4.5
0.9
6
1.2
2) Find an example from one of your GCSE courses which involves a relationship which
is linear.
 A Resource for Free-standing Mathematics Units
Karen Pittaway
Kingston College Mathematics Division