The Language of Functions and Graphs
Masters for Photocopying
CONTENTS
Examination
Questions
The journey
4
(12)*
Camping
5
(20)
Going to school
6
(28)
The vending machine
8
(38)
The hurdles race
9
(42)
The cassette tape
10
(46)
Filling a swimming pool
12
(52)
14
(64)
A2 Are graphs just pictures?
16
(74)
A3 Sketching graphs from words
18
(82)
A4 Sketching graphs from pictures
20
(88)
A5 Looking at gradients
22
(94)
24
(100)
Sketching graphs from words
26
(102)
Sketching graphs from pictures
28
(104)
t B 1 Sketching graphs from tables
32
(110)
B2 Finding functions in situations
35
(116)
B3 Looking at exponential functions
37
(12.0)
B4 A Function with several variables
39
(126)
Finding functions in situations
41
(131)
Finding functions in tables of data
43
(138)
45
(146)
Classroom Materials
t Al Interpreting points
Unit A
Supplementary
Interpreting
Unit B
Supplementary
A Problem
booklets:
points
booklets:
Collection
Problems:
Designing a water tank
1
A Problem
Collection
(cont'd)
The point of no return
47
(150)
'Warmsnug'
49
(154)
Producing a magazine
51
(158)
The Ffestiniog railway
53
(164)
Carbon dating
58
(170)
Designing a can
60
(174)
62
(178)
64
(182)
Feelings
69
(191)
The traffic survey
70
(192)
The motorway journey
71
(193)
Growth curves
72
(194)
Road accident statistics
73
(195)
The harbour tide
74
(196)
Alcohol
76
(198)
double glazing
Manufacturing
a computer
The missing planet
Graphs and other data for interpretation:
Support Material
A suggested programme
of meetings
81
6 unmarked scripts for
"Traffic"-An
'The hurdles race'
83
(237)
Marking record form
86
(236)
Approach to Distance-Time Graphs
t T1 Taking photographs from a helicopter
T2 From photographs
88
to cine film
90
T3, T4 More traffic problems
92
T5, T6, T7 Acceleration
96
"Snapshot.blanks"
Distance-time
and deceleration
master
102
grid master
103
* The numbers in brackets refer to the corresponding pages in the Module books.
t The masters for materials prefixed with an A, B or T should be used to form four
paged booklets,
by photocopying
©Shell Centre for Mathematical
Note:
back to back and folding in half.
Education,
University of Nottingham,
1985.
We welcome the duplication of the materials in this package for use
exclusively within the purchasing school or other institution.
2
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Examination
Questions
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THE JOURNEY
The map and the graph below describe a car journey
Crawley using the Ml and M23 motorways.
NOITINGHAM
from Nottingham
to
F
150
/
..JI.7E
~7
0 l/
Distance
(miles)
/
f/
100
J
B
lc
I
50
/
A
J
V
/
/
I
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2
3
Time (hours)
(i) Describe each stage of the journey, making use of the graph and the map. In
particular describe and explain what is happening from A to B; B to C;
C to D; D to E and E to F.
(ii) Using the information given above, sketch a graph to show how the speed of
the car varies during the journey.
80
60
Speed
(mph)
40
20
o
2
3
Time (hours)
©Shell Centre for Mathematical
Education,
University of Nottingham,
4 (12)
1985.
CAMPING
On their arrival at a campsite, a group of campers are given a piece of string 50
metres long and four, flag poles with which they have to mark out a rectangular
boundary for their tent.
They decide to pitch their tent next to a river as shown below ..This means that
the string has to be used for only three sides of the boundary.
(i)
If they decide to make the width of the boundary 20 metres, what will the
length of the boundary be?
(ii)
Describe in words, as fully as possible, how the length of the boundary
changes as the width increases through all possible values. (Consider both
small and large values of the width.)
(iii)
Find the area enclosed by the boundary for a width of 20 metres and for
some other different widths.
(iv)
Draw a sketch graph to show how the area enclosed changes as the width
of the boundary increases through all possible values. (Consider both
small and large values of the width.)
Area
Enclosed
Width of the boundary
The campers are interested in finding out what the length and the width of the
boundary should be to obtain the greatest possible area.
(v)
Describe,
width.
in words, a method by which you could find this length and
(vi)
Use the method
width.
©Shell Centre for Mathematical
you have described in part (v) to find this length and
Education,
University of Nottingham,
5 (20)
1985.
GOING TO SCHOOL
o
I
2
3
I
I
Graham
Scale (miles)
•
•
Jane
Paul
•
Peter
Jane, Graham, Susan, Paul and Peter all travel to school along the same country
road every morning. Peter goes in his dad's car, Jane cycles and Susan walks.
The other two children vary how they travel from day to day. The map above
shows where each person lives.
The following graph describes each pupil's journey to school last Monday.
6
Length
of
journey
to school
(miles)
•
•
•
4
2
•
•
1
o
20
40
Time taken to travel to school (minutes)
i)
Label each point on the graph with the name of the person it represents.
ii)
How did Paul and Graham travel to school on Monday?
iii)
Describe
_
how you arrived at your answer to part (ii) --
_
(continued)
©Shell Centre for Mathematical
Education,
University of Nottingham,
6 (28)
1985.
THE VENDING MACHINE
A factory cafeteria contains a vending machine which sells drinks.
On a typical day:
* the machine starts half full.
* no drinks are sold before 9 am or after 5 pm.
* drinks are sold at a slow rate throughout the day, except during the
morning and lunch breaks (10.30-11 am and 1-2 pm) when there is
greater demand.
* the machine is filled up just before the lunch break. (It takes about 10
minutes to fill).
Sketch a graph to show how the number of drinks in the machine might vary
from 8 am to 6 pm.
I
I
, I
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I
, 1 I •
11
I
I
Number of
drinks
in the machine
I
8
k
9
10
11
mornIng
12
1
2
)/(
3
4
5
afternoon
)/
Time of day·
©Shell Centre for Mathematical
Education,
University of Nottingham,
8 (38)
6
1985.
r-.---------------------------------------,I
GOING TO SCHOOL (continued)
iv)
Peter's father is able to drive at 30 mph on the straight sections of the road,
but he has to slow down for the corners. Sketch a graph on the axes below
to show how the car's speed varies along the route.
Peter ~s journey to School
I
30
Car's
Speed
(mph)
20
10-
o
I
1
2
4
Di~tance
from
Peter'~
home
I
i
5
6
(milt:s)
I
©Shell Centre for Mathematical
Education,
University of Nottingham,
7 (29)
1985.
)
THE HURDLES RACE
400
,/
./
/
;"
Distance
(metres)
,,"
/
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60
Time (seconds)
The rough sketch graph shown above describes what happens when 3 athletes
A, 8 and C enter a 400 metres hurdles race.
Imagine that you are the race commentator. Describe what is happening
carefully as you can. You do not need to measure anything accurately.
©Shell Centre for Mathematical
Education,
University of Nottingham,
9 (42)
1985.
as
THE CASSETTE TAPE
This diagram represents a cassette recorder just as it is beginning to playa tape.
The tape passes the "'head" (Labelled H) at a constant speed and the tape is
wound from the left hand spool on to the right hand spool.
At the beginning,
lasts 45 minutes.
(i)
the radius of the tape on the left hand spool is 2.5 cm. The tape
Sketch a graph to show how the length of the tape on the left hand spool
changes with time.
Length of tape
on left hand
spool
o
10
20
30
40
50
Time (minutes)
(continued)
©Shell Centre for Mathematical
Education,
University of Nottingham,
10 (46)
1985.
THE CASSETTE TAPE (continued)
(ii)
Sketch a graph to show how the radius of the tape on the left hand spool
changes with time.
lro.
3Radius of tape
on left hand
spool (cm)
21 -
o
I
I
10
20
I
30
I
I
40
50
"-
/'
Time (minutes)
(iii)
Describe and explain how the radius of the tape on the right-hand spool
changes with time.
©Shell Centre for Mathematical
Education,
University of Nottingham,
11 (47)
1985.
FILLING A SWIMMING POOL
(i)
A rectangular
swimming pool is being filled using a hosepipe which
delivers water at a constant rate. A cross section of the pool is shown
below.
-_-_-_-_-_-
t
-----d
~
Describe fully, in words, how the depth (d) of water in the deep end of the
pool varies with time, from the moment that the empty pool begins to fill.
(ii)
A different
rectangular
pool is being filled in a similar way.
Sketch a graph to show how the depth (d) of water in the deep end of the
pool varies with time, from the moment that the empty pool begins to fill.
Assume that the pool takes thirty minutes to fill to the brim.
2
Depth of water
in the pool (d)
(metres)
1
o
©Shell Centre for Mathematical
10
20
Time (minutes)
Education,
30
University of Nottingham,
12 (52)
1985.
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HI. (contd)
SOME HINTS ON SKETCHING
GRAPHS FROM TABLES
Look again at the balloon problem, "How far can you see?"
The following discussion should help you to see how you can go about sketching
quick graphs from tables without having to spend a long time plotting points.
*
As the balloon's'height increases by equal amounts, what happens to the 'distance
to the horizon'? Does it increase or decrease?
Balloon's height (m)
Distance to horizon (km)
5
8
10 20 30 40 50 100 500 1000
11 16 20 23 25 36 80 112
Does this distance
increase
by equal amounts? ...
10
20
Balloon's
30
40
height
... or increase by greater
and greater amounts?
..
10
20
Balloon's
30
40
height
... or increase by smaller and
smaller amounts?
Now ask yourself:
• do the other numbers in the table
fit in with this overall trend?
• will the graph cross the axes?
If so, where?
©Shell Centre for Mathematical
Education,
10
20
Balloon's
University of Nottingham,
34 (Ill)
1985.
30
40
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44 (138)
DESIGNING
2 metres
A WATER TANK
:>
/
/
2 metres
A square metal sheet (2 metres by 2 metres) is to be made into an open-topped
water tank by cutting squares from the four corners of the sheet, and bending the
four remaining rectangular pieces up, to form the sides of the tank. These edges
will then be welded together.
*
How will the final volume of the tank depend upon the size of the squares cut
from the corners?
Describe
*
your answer by:
a)
Sketching
a rough graph
b)
Explaining
c)
Trying to find an algebraic formula
the shape of your graph in words
How large should the four corners be cut, so that the resulting volume of the
tank is as large as possible?
©Shell Centre for Mathematical
Education,
University of Nottingham,
45 (146)
1985.
DESIGNING
*
A WATERTANK
... SOME HINTS
Imagine cutting very small squares from the corners of the metal sheet. In
your mind, fold the sheet up. Will the resulting volume be large or small?
Why?
Now imagine cutting out larger and larger squares ....
What are the largest squares you can cut? What will the resulting volume be?
*
Sketch a rough graph to describe your thoughts and explain it fully in words
underneath:
Volume of
the
tank (m3)
,
/
Length of the sides of the squares (m).
*
In order to find a formula, imagine cutting a square x metres by x rnetres
from each corner of the sheet. Find an expression for the resulting volume.
*
Now try plotting an accurate graph.
(A suitable scale is 1 cm represents 0.1 metres on the horizontal axis, and
1 cm represents 0.1 cubic metres on the vertical axis).
How good was your sketch?
*
Use your graph to find out how large the four corner squares should be cut,
so that the resulting volume is maximised.
©Shell Centre for Mathematical
Education,
University of Nottingham,
46 (147)
1985.
THE POINT OF NO RETURN
I magine that you are the pilot of the light aircraft in the picture, which is capable
of cruising at a steady speed of 300 km/h in still air. You have enough fuel on
board to last four hours.
You take off from the airfield and, on the outward journey,
a 50 km/h wind which increases your cruising speed relative
are helped along by
to the ground to 350
km/h.
Suddenly
you realise that on your return journey
and will therefore
slow down to 250 km/h.
you will be tlying into the wind
What is the maximum distance that you can travel from the airfield, and still
be sure that you have enough fuel left to make a safe return journey?
:;:
Investigate
these 'points
©Shell Centre for Mathematical
of no return'
Education,
for different
wind speeds.
University of Nottingham,
47 (150)
1985.
THE POINT OF NO RETURN ... SOME HINTS
*
Draw a graph to show how your distance from the airfield will vary with time.
How can you show an outward speed of 350 km/h?
How can you show a return speed of 250 km/h?
800
--
700
E
600
rJl
<l)
•...
<l)
l-.
..Q
~-...-
:g
500
<l)
t;::
l-.
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l-.
4-<
300
<l)
u
t: 200
ro
•...
rJ'J
Cl
100
o
2
Time (hours)
4
3
Use your graph to find the maximum distance you can travel from the
airfield, and the time at which you should turn round.
*
On the same graph, investigate the 'points of no return' for different wind
speeds. What kind of pattern do these points make on the graph paper?
Cari you explain why?
Suppose
the windspeed is w km/h,
the 'point of no return' is d km from the airfield
and the time at which you should turn round is t hours.
Write down two expressions for the outward
one involving wand one involving d and t.
Write down two expressions for the homeward
one involving wand one involving d and t.
Try to express d in terms of only
equations.
t,
speed
of the aircraft,
speed of the aircraft,
by eliminating w from the two resulting
Does this explain the pattern made by your 'points of no return '?
©Shell Centre for Mathematical
Education,
University of Nottingham,
48 (151)
1985.
"W ARMSNUG DOUBLE GLAZING"
(The windows on this sheet are all
drawn to scale: 1 em represents 1 foot).
A
How have "Warmsnug" arrived at
the prices shown on these windows?
Which window has been given
an incorr~ct price? How much
should it cost?
Explain your reasoning clearly.
£66
r
J
£46
£55
~_________
~....
----~J ..
©Shell Centre for Mathematical
Education,
£84
I
University of Nottingham,
49 (154)
1985.
"W ARMSNUG"
*
DOUBLE GLAZING ... SOME HINTS
Write down a list of factors which may affect the price that "Warmsnug" ask
for any particular window:
e.g.
Perimeter,
Area of glass needed,
*
U sing your list, examine the pictures of the windows in a systematic manner.
*
Draw up a table, showing all the data which you think may be relevant.
(Can you share this work out among other members of your group?)
*
Which factors or combinations of factors is the most important in
determining the price?
Draw scattergraphs to test your ideas. For example, if you think that the
perimeter is the most important factor, you could draw a graph showing:
Cost
of
window
"-
Perimeter of window
"
*
Does your graph confirm your ideas? If not, you may have to look at some
other factors.
*
Try to find a point which does not follow the general trend on your graph.
Has this window been incorrectly priced?
*
Try to find a formula which fits your graph, and which can be used to predict
the price of any window from its dimensions.
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
50 (155)
PRODUCING
A MAGAZINE
A group of bored, penniless teenagers want to make some money by producing
and selling their own home-made magazine. A sympathetic teacher offers to
supply duplicating facilities and paper free of charge, at least for the first few
Issues.
a) Make a list of all the important decisions they must make.
Here are three to start you off:
How long should t~e magazine be?
(I pages)
How many writers will be needed?
(w writers)
How long will it take to write?
(t hours)
b) Many items in your list will depend on other items.
For example,
For a fixed number of people involved,
the longer the magazine, the longer
it will take to write.
I~
For a fixed length of magazine,
the more writers there are,
Complete the statement,
to illustrate it.
and sketch a graph
w writers
"-
'"
Write down other relationships you can find,
and sketch graphs in each case.
2
The group eventually decides to find out how many potential customers
there are within the school, by producing a sample magazine and conducting
a survey of 100 pupils, asking them "Up to how much would you be prepared
to pay for this magazine?" Their data is shown below:
Nothing
Selling price (5 pence)
Number
prepared
to pay this price (n people)
100
10 20
30
40
82
40
18
58
How much should they sell the magazine for in order to maximise their
profit?
3
After a f~w issues, the teacher decides that he will have to charge the pupils
lOp per magazine for paper and duplicating.
How much should they sell the magazineJor
©Shell Centre for Mathematical
Education,
now?
University of Nottingham,
51 (158)
1985.
PRODUCING A MAGAZINE ... SOME HINTS
1
2
3
Here is a more complete list of the important factors that must be taken into
account:
Who is the magazine for?
(schoolfriends?)
What should it be about?
(news, sport, puzzles, jokes .. ?)
How long should it be?
(l pages)
How many writers will it need?
(w writers)
How long will it take to write?
(t hours)
How many people will buy it?
(n people)
What should we fix the selling
price at?
(s pence)
How much profit will we make
altogether?
(p pence)
How much should we spend on
advertising?
(a pence)
*
Can you think of any important factors that are still missing?
*
Sketch graphs to show how: t depends on w; w depends on I;
n depends on s;- p depends on s; n depends on a.
*
Explain the shape of each of your graphs in words.
*
Draw a graph of the information
*
Explain the shape of the graph.
*
What kind of relationship is this?
(Can you find an approximate formula which relates n to s?)
*
From this data, draw up a table of values and a graph to show how the
profit (p pence) depends on the selling price (s pence).
(Can you find a formula which relates p and s?)
*
Use your graph to find the selling price which maximises the profit made.
given in the table of data.
Each magazine costs lap to produce.
*
Suppose we fix the selling price at 20p.
How many people will buy the magazine? How much money will he
raised by selling the magazine, (the 'revenue')? How much will these
magazines cost to produce? How much actual profit will therefore be
made?
*
Draw up a table of data which shows how the revenue, production costs
and profit all vary with the selling price of the magazine.
*
Draw a graph from your table and use it to decide on the best selling price
for the magazine.
©Shell Centre for Mathematical
Education,
University of Nottingham,
52 (159)
1985.
THE FFESTINIOG RAILWAY
This railway line is one of the most famous in Wales.
Your task will be to devise a workable timetable for
running this line during the peak tourist season.
The following facts will need to be taken into account:;;: There are 6 main stations along the 131/2 mile track:
(The distances
Blaenau
Ffestiniog
between them are shown in miles)
Z ..
Tan-y-Bwlch
P/~
~
2
Three steam trains are to operate a shuttle service. This means that they will
travel back and forth along the line from Porthmadog to Blaenau Ffestiniog with
a IO-minute stop at each end. (This should provide enough time for drivers to
change etc.)
*
The three trains must start and finish each day at Porthmadog.
The line is single-track. This means that trains cannot pass each other, except at
specially designed passing places. (You will need to say where these will be
needed. You should try to use as few passing places as possible.)
Trains should depart from stations at regular intervals if possible.
The journey from Porthmadog to Blaenau Ffestiniog is 65 minutes (includirig
stops at intermediate stations. These stops ar~ very short and may be neglected in
the timetabling).
*
The first train of the day will leave Porthmadog
*
The last train must return to Porthmadog by 5.00 p.m. (These times are more
restricted than .those that do, in fact, operate.)
©Shell Centre for Mathematical
Education,
at 9.00 a.m.
University of Nottingham,
53 (164)
1985.
THE "FFESTINIOG RAILWAY" ... SOME HINTS
Use a copy of -the graph paper provided to draw a distance-time
9.00 a.m. train leaving Porthmadog.
graph for the
Try to show, accurately:
•
•
•
•
The
The
The
The
outward journey from Porthmadog to Blaenau Ffestiniog.
waiting time at Blaenau Ffestiniog.
return journey from Blaenau Ffestiniog to Porthmadog.
waiting time at Porthmadog ... and so on.
What is the interval between departure times from Porthmadog for the above
train '?
How can we space the two other trains regularly between these departure times?
Draw similar graphs for the other two trains.
How many passing places are needed'? Where will these have to be?
From your graph, complete the following timetable:
Miles
Station
Daily Timetable
0
Porthmadog
d
2
Minffordd
d
31/~
Penrhyn
d
7112
Tan-y-Bwlch
d
121j~
Tanygrisiau
d
13112
Blaenau Ffestiniog
a
0
Blaenau Ffestiniog
d
11/~
Tanygrisiau
d
6
Tan-y-Bwlch
d
101/4
Penrhyn
d
11112
Minffordd
d
13112
Porthmadog
a
09.00
Ask your teacher for a copy of the real timetable,
compares with your own.
©Shell Centre for Mathematical
Education,
and write about how it
University of Nottingham,
54 (165)
1985.
o
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©Shell Centre for Mathematical Education, University of Nottingham, 1985.
55 (166)
-
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::lEa:
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©Shell Centre for Mathematical Education, University of Nottingham, 1985.
56 (167)
Monday 15 July to Friday 30 August
(Also Spring Holiday Week Sunday 26 May to Thursday 30 May)
MONDAYS TO THURSDAYS -------~-
-_._--,--_._---_._~-----------~-
d.• :'.
1155
1237
0955
1040
Minffordd
iPenrhyn
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i BI. Ffesliniog a.
0959
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1025
1043
1055
-~---
_::=-===---::-::--=:=
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Flestlnlol1
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I nd dn
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Llendudno Jcn.
BI. Flestlnlol1
b
rs:an~~~r;~O;}
en"sh
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onwy Valley Ime
i
SERVICE
f.-
1330
1431
1441
8.
8rl/lsh
Rail
Coast
a.
1105 1155
1111 1201
1133 1225
1149 1245
1154.1250
Mlnltordd
Ba,mouth
M/nlfordd
Pwllhell
Ime
SERVICE
1319 1409
1325 1415
1345 1435
1407 1458
1417
1507
------- - ---
d.
1340
1346
1405
1424
1429
a.
1710
1751
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1633
1504
1510
1530
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1433
1514
a.
d:
1430
1436
1500
1520
1525
1500
1530
1520
1606
1433
1625
1726
1736
1333
1343
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1245
1251
1315
1335
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1400
1343
1422
1247$0
1325$0
1504 1554 1644
1510 1600 1650
1530 1620 1710
1548 1641 1728
1600
1652 1740
----_._--_._-------
1450
1548
1558
1110
1721
1224
raCFfestiniog
Tanygrisiau
i Tan-y-Bwlch
:Penrhyn
, Minffordd
ISe:on~~~g~~wl
1229
1235
1255
1313
1325
d.
e.
d.
d.
I
Cambn,m
1044 1134
1050 1140
1112 1205
1135 1223
1146
1233
-----~------
1545 1635
1400
1433
1343
1422
1135
1205
0938
1008
~-
----
0950 1035 1125 1220 1310 14001455
1615
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1659
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1706
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1110
1121
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• 1340
1955
2036
--------
1834
1901
1955
2036
2239
• 1209 1304 1354 1444 1539 1629 1719 1805 1848
DAILY
Monday 16 Sept.
Saturday 31 August to Sunday 15 September
MONDAYS TO FRIDAYS
1125 1~400
i
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d.
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1135
a.
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1205
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denotes DIESEL HAULED
TRAINS - 'ECONOMY'
FARES AVAILABLE.
: Considerable reductions
I for passengers commencing a FULL RETURN
JOURNEY by such trains.
The return journey may
be made by any train.
All other services are
normally steam hauled.
RQ Stops on request only.
Passengers
wishing
to
alight should Inlorm the
guard before bOlrdlng.
Passengers
wishing
to
Join should give a clear
•I
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©Shell Centre for Mathematical
Education,
Please apply to:
FFESTINIOG RAILWAY
HARBOUR STATION
PORTH MADOG
GWYNEDD
Telephone:
PORTHMADOG
(0766) 2340/2384
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be made
to ensure
running as timetable but
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not guarantee advertised
connections
nor the
advertised traction in the
event of breakdown or
other
obslruclion
of
services.
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SATS. & SUNS. to Sunday 3 Nov.
1455 1635:0950 1220 _145509501220
University of Nottingham,
57 (16S)
1985.
A special service will operate on
21 and 22 December. Details
, available from 1 Oclober.
Father Christmas will meet the
trains and distribute presents to
the children.
All seats reservable - Advance
Booking Essential.
CARBON DATING
Carbon dating is a technique for discovering
the age of an ancient object, (such as a bone
or a piece of furniture) by measuring the
amount of Carbon 14 that it contains.
While plants and 'animals are alive, their
Carbon 14 content remains constant, but when
they die it decreases to radioactive decay.
The amount, a, of Carbon 14 in an object
t thousand years after it dies is given by
the formula:
a
= 15.3 x- 0.886
t
(The quantity ""a" measures the rate of Carbon 14 atom disintegrations and this
is measured in "counts per minute per gram of carbon (cpm)")
1
Imagine that you have two samples of wood. One was taken from a fresh tree
and the other was taken from a charcoal sample found at Stonehenge and is
4000 years old.
How much Carbon 14 does each sample contain? (Answer in cpm's)
How long does it take for the amount of Carbon 14 in each sample to be
halved?
These two answers should be the same, (Why?) and this is called the half-life
of Carbon 14.
2
Charcoal from the famous Lascaux Cave in France gave a count of2.34 cpm.
Estimate the date of formation of the charcoal and give a date to the
paintings found in the cave.
3
Bones A and B are x and y thousand years old respectively. Bone A contains
three times as much Carbon 14 as bone B.
What can you say about x and y?
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
58 (170)
CARBON DATING ... SOME HINTS
U sing a calculator, draw a table of values and plot a graph to show how the
amount of Carbon 14 in an object varies with time.
t
0
(1000's of years)
1
3
2
4
10 ... 17
5
6
7
8
9
9
10
11
12
13 14
a (c.p.m)
15
~
II
----8
0.0
0..
u
"-'
.•...
u
(])
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10
0
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0
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~
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1
2
3
4
5
6
7
8
Age of object (in 1000's of years) = t
Use your graph to read off answers to the questions.
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
59(171)
15
16
17
DESIGNING A CAN
A cylindrical can, able to contain half a litre of drink, is to be manufactured
aluminium. The volume of the can must therefore be 500 cm3•
*
Find·the radius and height of the can which will use the least aluminium, and
therefore be the cheapest to manufacture. (i.e., find out how to minimise the
surface area of the can).
State clearly any assumptions
*
from
you make.
What shape is your can? Do you know of any cans that are made with this
shape? Can you think of any practical reasons why more cans are not this
shape?
©Shell Centre for Mathematical
Education,
University of Nottingham,
60 (174)
1985.
DESIGNING A CAN ... SOME HINTS
*
You are told that the volume of the can must be 500 cm3•
If you made the can very tall, would it have to be narrow or wide? Why?
If you made the can very wide, would it have to be tall or short? Why?
Sketch a rough graph to.describe how the height and radius of the can have to
be related to each other.
*
Let the radius of the can be r cm, and the height be h cm.
Write down algebraic expressions which give
-
the volume of the can
-
the total surface area of the can, in terms of rand h.
(remember to include the two ends!).
*
U sing the fact that the volume of the can must be 500 cm3, you could
either: - try to find some possible pairs of values for rand h
(do this systematically if you can).
- for each of your pairs, find out the corresponding surface area.
or: - try to write one single expression for the surface area in terms of r,
by eliminating h from your equations.
*
Now plot a graph to show how the surface area varies as r is increased, and.
use your graph to find the value of r that minimises this surface area.
*
Use your value of r to find the corresponding value of h. What do you notice
about your answers? What shape is the can?
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
61 (175)
MANUFACTURING
A COMPUTER
Imagine that you are running a small business which assembles and sells two
kinds of computer: Model A and Model B (the cheaper version). You are only
able to manufacture up to 360 computers, of either type, in any given week.
The following ta,ble shows all the relevant data concerning the employees at
your company:
Job Title
Numper of people
doing this job
Assembler
100
Inspector
4
Job description
Pay
Hours
worked
This job involves
putting the computers
together
£100
per week
36 hours
per week
This job involves
testing and
correcting any
faults in the
computers before
they are sold
£120
per week
35 hours
per week
The next table shows all the relevant data concerning the manufacture of the
computers.
Model A
ModelB
Total assembly time in man-hours
for each computer
12
6
Total inspection and correction time
in man-minutes for each computer
10
30
Component costs for each computer
£80
£64
Selling price for each computer
£120
£88
At the moment, you are manufacturing and selling 100 of Model A and 200
of Model B each week.
*
What profit are you making at the moment?
*
How many of each computer should you make in order to improve this
worrying situation?
*
Would it help if you were to make some employees redundant?
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
62 (178)
MANUF ACTURING A COMPUTER
1
2
Suppose you manufacture
100 Model A's and 200 Model B's in one week:
*
How much do you pay in wages?
*
How muc~ do you pay for components?
*
What is your weekly income?
*
What profit do you make?
N ow suppose
each week.
*
... SOME HINTS
that you manufacture
x Model A and y Model B computers
Write down 3 inequalities involving x and y. These will include:
- considering the time it takes to assemble the computers, and the total
time that the assemblers have available.
- considering the time it takes to inspect and correct faults in the
computers, and the total time the inspectors have available.
Draw a graph and show the region satisfied by all 3 inequalities:
Number of
Model B
computers
manufactured
(y)
a
Number
100
200
300
400
of Model A computers manufactured
(x)
3
Work out an expression which tells you the profit made on x Model A and
y Model B computers.
4
Which points on your graph maximise your profit?
©Shell Centre for Mathematical
Education,
University of Nottingham,
63 (179)
1985.
THE MISSING PLANET 1.
In our solar system, there are nine major planets, and many other smaller bodies
such as comets and meteorites. The five planets nearest to the sun are shown in
the diagram below.
.. .- .
: ~'::·).~i".
•
Mars
~.
'Ii
.. ::;:,,~".Asteroids
... .
f
•••
•
~~':.'
.
"~'.
:-
• ••••
••
.. .
#
·:.:xr;'
. .:.-.;.'
,.......
.
•
"'.I.
,".
•
'
...~~.~<~;'
.. ~"'
Jupiter
Between Mars and Jupiter lies a belt of rock fragments called the 'asteroids'.
These are, perhaps, the remains of a tenth planet which disintegrated many
years ago. We shall call this, planet 'X'. In these worksheets, you will try to
discover everything you can about planet 'X' by looking at patterns which occur
in the other nine planets.
How far was planet 'X' from the sun, before it disintegrated?
The table below compares the distances of some planets from the Sun with that
of our Earth. (So, for example, Saturn is 10 times as far away from the Sun as the
Earth. Scientists usually write this as 10 A.U. or 10' Astronomical Units').
*
*
*
*
Can you spot any pattern in
the sequence of approximate
relative distances.
Can you use this pattern to
predict the missing figures?
So how far away do you think
planet 'X' was from the Sun?
(The Earth is 93 million miles
away)
Check your completed table
with the planetary data
sheet.
Where does the pattern seem
to break down?
Planet
Mercury
Venus
Earth
Mars
Planet X
Jupiter
Saturn
Uranus
Neptune
Pluto
Relative Distance from Sun, approx
(exact figures are shown in brackets)
?
0.7
1
1.6
?
5.2
10
19.6
?
?
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
64 (182)
(0.72)
(1)
(1.52)
(5.20)
(9.54)
(19.18)
'-
o
r
1-0
r/J
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S
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8
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oo
~
00
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t"'-
oo
00
©Shell Centre for Mathematical
Education,
University of Nottingham,
65 (183)
1985.
THE MISSING PLANET ...
SOME BACKGROUND INFORMATION
In 1772, when planetary distances were still only known in relative terms, a
German astronomer named David Titius discovered the same pattern as the one
you have been looking at. This 'law' was publisheQ by Johann Bode in 1778 and
is now commonly known as "Bode's Law". Bode used the pattern, as you have
done, to predict the existence of a planet 2.8 AU from the sun. (2.8 times as far
away from the Sun as the Earth) and towards the end of the eighteenth century
scientists began to search systematically for it. This search was fruitless until
New Year's Day 1801, when the Italian astronomer Guiseppe Piazzi discovered
a very small asteroid which he named Ceres at a distance 2.76 AU from the
Sun-astonishingly
close to that predicted by Bode's Law. (Since that time,
thousands of other small asteroids have been discovered, at distances between
2.2 and 3.2 AU from the sun.)
In 1781, Bode's Law was again apparently confirmed, when William Herschel
discovered the planet Uranus, orbitting the sun at a distance of 19.2 AU, again
startlingly close to 19.6 AU as predicted by Bode's Law. Encouraged by this,
other astronomers used the 'law' as a starting point in the search for other distant
planets.
However, when Neptune and Pluto were finally discovered, at 30 AU and 39
A U from the Sun, respectively, it was realised that despite its past usefulness,
Bode's 'law' does not really govern the design of the solar system.
©Shell Centre for Mathematical
Education,
University of Nottingham,
66 (1~4)
1985.
THE MISSING PLANET 2.
Look at the Planetary
data sheet, which contains 7 statistics for each planet.
The following scientists are making hypotheses about the relationship between
these statistics:
A
@
The further a
planet is away from
the Sun, the longer
it takes to orbit
the Sun.
C
B
Do you agree with these hypotheses?
sheet)
*
e
G
ct
•••
J
The smaller
the planet
the slower it
spIns.
How true are they? (Use the data
Invent a list of your own hypotheses.
Sketch a graph to illustrate each of them.
One way to test a hypothesis is to draw a scattergraph. This will give you some
idea of how strong the relationship is between the two variables.
For example,
here is a 'sketch' scattergraph
C)
c::
::s
.•...
0.
Cl)
Z
testing the hypothesis of scientist A:
x
o
.•...
::s
0:
X
.~
.n
1-0
o
o
.•...
We can therefore predict the orbital
time for Planet X. It should lie
between that of Mars (2 years) and
Jupiter (12 years). (A more accurate
statement would need a more accurate
graph. )
.
X
.••••
Cl)
m
1-0
Cl)
.••••
c::
.ro 0..
1-0,_
ro ~~
Notice that:
There does appear to be a
relationship between the distance
a planet is from the Sun and the
time it takes to orbit once. The
hypothesis seems to be confirmed.
::s
~
~
X ? X
Distance from sun
•
Sketch scattergraphs to test your own hypotheses.
out about Planet X? What cannot be found?
©Shell Centre for Mathematical
Education,
What else can be found
University of Nottingham,
67(185)
1985.
THE MISSING PLANET 3.
After many years of observation the famous
mathematician
Johann Kepler (1571-1630)
found that the time taken for a planet to orbit
the Sun (T years) and its average distance
from the Sun (R'miles) are related by the
formula
where K is a constant value.
*
Use a calculator to check this formula from the data sheet, and find the value
ofK.
Use your value of K to find a more accurate estimate for the orbital time (1)
of Planet X. (You found the value of R for Planet X on the first of these
sheets).
*
We asserted that the orbits of planets are 'nearly circular'. Assuming this is
so, can you find another formula which connects
-
The average distance of the planet from the Sun (R miles)
-
The time for one orbit (T years)
-
The speed at which the planet 'flies through space' (V miles per hour)?
planet
(Hint: Find out how far the planet
moves during one orbit. You can
write this down in two different
ways using R, T and V)
(Warning: Tis in years, Vis in
miles per hfJur)
Use a calculator to check your formula from the data sheet.
Use your formula, together with what you already know about Rand
find a more accurate estimate for the speed of Planet X.
*
Assuming
connecting
that
the planets
are spherical,
T, to
can you find a relationship
-
The diameter of a planet (d miles)
-
The speed at which a point on the equator spins (v miles
per hour)
-
The time the planet takes to spin round once (t hours)?
Check your formula from the data sheet.
©Shell Centre for Mathematical
Education,
University of Nottingham,
68 (186)
1985.
FEELINGS
These graphs show how a girl's feelings varied
during a typical day.
Her timetable
for the day was as follows:
7 .()() am
woke up
went to school
Assembly
Science
Break
Maths
Lunchtime
Games
Break
French
went home
did homework
went lO-pin bowling
went to bed
am
l) .()() am
l).3() am
1O.3() am
I I.()() am
12 .()() am
I .3() pm
2ASpm
X.()()
.3.()()
.+.()()
pm
pm
h. ()() pm
7 .()() pm
I ().3() pm
Happy
(a)
Try to explain the shape
of each graph, as fully as
possible.
(b)
How many meals did she eat?
Which meal was the biggest?
Did she eat at breaktimes?
How long did she spend
eating lunch?
Which lesson did she enjoy
the most?
When was she ""tired and
depressed?" Why was this?
When was she ""hungry but
happy?" Why was this?
'"0
o
o
~
Sad
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11
Time of day
Full of
energy
;>,
OJ)
~
C)
c:
U.J
Exhau~teJ
I
I
I
I
I
I
I
I
I
I
I
6 7 8 9 10 11 12 1 2
I
3 4 5
Time of day
I
I
,
I
I
I
6 7 8 9 1011
Make up some more questions
like these, and give them to
your neighbour to solve.
Full
up
~
C)
OJ)
f
c:
::l
:r:
Starving
,
I
I
I
6 7 8
<)
I
I
I
I
t
I
1
I
I
I
I
:2 3 4
Time of day
6
7
8
<)
1() 11 12 I
©Shell Centre for Mathematical
Education,
I
(c)
I
1() 11
Sketch graphs to show how
your feelings change during
the day. See if your
neighbour can interpret them
correctly.
University of Nottingham,
6Y ( 1Y I)
1Y~5.
THE TRAFFIC SURVEY
A survey was conducted to discover the
volume of traffic using a particular
road. The results were published in
the form of the graph which shows the
number of cars using the road at any
specified time during a typical
Sunday and Monday in June.
1.
Try to explain, as fully as possible, the shape of the graph.
2.
Compare Sunday's graph with Monday's. What is suprising?
3.
Where do you think this road could be? (Give an example of a road you
know of, which may produce such a graph.)
900
800
"0
~
0
•...
700
(1)
...c:
~
c=
0
600
.--
500
rJJ
(1)
u
...c:
(1)
>
4-1
0
~
400
(1)
..0
e;:j
300
Z
200
100
0
2
t
Midnight
I~
4
6
8
10 12 2
t
4
Noon
Sunday --
6
8
10 12 2
Midnight
10 12 2
t
Noon
~
Monday
t
4
6
8
4
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
70 (192)
6
8
10 12
f
Midnight
~
THE MOTORW AY JOURNEY
8
7
---en
c::
..9
~
en
'-'
~
~
u
~
E
f' ~
I'
6
I"""-
5
"
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~ 3
~
" ,
"
•..•..••...
""-
(l)
2
" '~
f'
....•..•...
•...
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•..•..••...
4
.5
~
'-
.....•.•...
~
.....•.•...
~
.....•.•...
~
1
a
50
100
150
200
250
Distance travelled in miles
The above graph shows how the amount of petrol in my car varied during a
motorway journey.
Write a paragraph to explain the shape of the graph. In particular answer the
following questions:
1
How much petrol did I have in my tank after 130 miles?
2
My tank holds about 9 gallons. Where was it more than half full?
3
How many petrol stations did I stop at?
4
At which station did I buy the most petrol? How can you tell?
5
If I had not stopped anywhere, where would I have run out of petrol?
6
If I had only stopped once for petrol, where would I have run out?
7
How much petrol did I use for the first 100 miles?
8
How much petrol did I use over the entire journey?
9
How many miles per gallon (mpg) did my car do on this motorway?
I left the motorway, after 260 miles, I drove along country roads for 40 miles and
then 10 miles through a city, where I had to keep stopping and starting. Along
country roads, my car does about 30 mpg, but in the city irs more like 20 mpg.
10
Sketch a graph to show the remainder of my journey.
©Shell Centre for Mathematical
Education,
University of Nottingham,
71 (193)
1985.
GROWTH CURVES
Paul and Susan are two fairly typical people. The following graphs compare how
their weights have changed during their first twenty years.
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Age in years
Write a paragraph
comparing the shape of the two graphs.
everything you think is important.
Write down
Now answer the following:
1
How much weight did each person put on during their "secondary
years (between the ages of 11 and 18)?
2
When did Paul weigh more than Susan? How can you tell?
3
When did they both weigh the same?
4
When was Susan putting on weight most rapidly?
school"
How can you tell this from the graph?
How fast was she growing at this time? (Answer in kg per year).
5
When was Paul growing most rapidly? How fast was he growing at this time?
6
Who was growing faster at the age of 14? How can you tell?
7
When was Paul growing faster than Susan?
8
Girls tend to have boyfriends older than themselves. Why do you think this is
so? What is the connection with the graph?
©Shell Centre for Mathematical
Education,
University of Nottingham,
72 ( 19-4)
1985.
ROAD ACCIDENT STATISTICS
The following four graphs show how the number of road accident casualties per
hour varies during a typical week.
Graph A shows the normal pattern
Thursday.
*
*
*
Which graphs correspond
for Monday,
Tuesday,
Wednesday
and
to Friday, Saturday and Sunday?
Explain the reasons for the shape of each graph, as fully as possible.
What e.vidence
accidents?
there to show that alcohol
IS
IS
a major cause of road
Graph B
Graph A
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THE HARBOUR TIDE
The graph overleaf, shows how the depth of water in a harbour varies on a
particular Wednesday.
1
Write a paragraph
which describes in detail what the graph is saying:
When is high/low tide? When is the water level rising/falling?
When is the water level rising/falling most rapidly?
How fast is it rising/falling at this time?
What is the average depth of the water? How much does the depth vary
from the average?
2
Ships can only enter the harbour when the water is deep enough. What
factors will determine when a particular boat can enter or leave the harbour?
The ship in the diagram below has a draught of 5 metres when loaded with
cargo and only 2 metres when unloaded.
Discuss when it can safely enter and leave the harbour.
Harbour
depth
Draught·····::· ...-::~·::.:::.:··.·.·
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Make a table showing when boats of different draughts can safely enter and
leave the harbour on Wednesday.
3
Try to complete the graph in order to predict how the tide will vary on
Thursday. How will the table you draw up in question 2 need to be adjusted
for Thursday? Friday? ...
4
Assuming
(Where
that the formula which fits this graph is of the form
d = A + B cos(28t + 166t
d = depth of water in metres
t = time in hours after midnight on Tuesday night)
Can you find out the values of A and B?
How can you do this without substituting in values for t?
©Shell Centre for Mathematical
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1985.
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ALCOHOL
Read through
questions:
the data sheet carefully, and then try to answer the following
U sing the chart and diagram on page 2, describe and compare the
effects of consuming different quantities of different drinks.
(eg: Compare the effect of drinking a pint of beer with a pint of whisky)
Note that 20 fl oz = 4 gills = 1 pint.
Illustrate
your answer with a table of some kind.
An 11 stone man leaves a party at about 2 am after drinking 5 pints of
beer. He takes a taxi home and goes to bed. Can he legally drive to
work at 7 am the next morning? When would you advise him that he is
fit to drive? Explain your reasoning as carefully as possible.
The five questions below will help you to compare and contrast the information
presented on the data sheet.
1
Using only the information presented in words by the "Which?" report,
draw an accurate
at 2 am.
2
graph showing the effect of drinking 5 pints of beer
a)
What will the blood alcohol level rise to?
b)
How long will it take to reach this level?
c)
How quickly will this level drop?
d)
What is the legal limit for car drivers? How long will this person remain
unfit to drive? Explain your reasoning.
Using only the formula provided,
draw another graph to show the effect of drinking 5 pints of beer.
How does this graph differ from the graph produced above?
Use your formula to answer la) b) c) d) again.
Compare
3
your answers with those already obtained.
Using only the table of data from the AA bC!0kof driving,
draw another graph to show the effect of an 11 stone man drinking 5 pints of
beer.
ComP':l~e this graph to those already obtained.
Answer
above.
1a) b) c) d) from this graph, and compare your answers with those
©Shell Centre for Mathematical
Education,
University of Nottingham,
76(198)
1985.
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ALCOHOL (continued)
4
Compare the graph taken from the Medical textbook with those drawn for
questions 1,2 and 3. Answer question la) b) c) and d) concerning the 11
stone man from this graph.
5
Compare the advantages and disadvantages of each mode of representation:
words, formula, graph and table, using the following criteria:
Compactness
Accuracy
(does it take up much room?)
(is the information over-simplified?)
Simplicity
Versatility
(is it easy to understand?)
(can it show the effects of drinking different
amounts of alcohol easily?)
(which set of data do you trust the most?
Why? Which set do you trust the least? Why?)
Reliability
A business woman drinks a glass of sherry, two glasses of table wine
and a double brandy during her lunch hour, from 1 pm to 2 pm. Three
hours later, she leaves work and joins some friends for a meal, where
she drinks two double whiskies.
Draw a graph to show how her blood/alcohol level varied during the
entire afternoon (from noon to midnight). When would you have
advised her that she was unfit to drive?
©Shell Centre for Mathematical Education, University of Nottingham, 1985.
79 (200)
Support
Materials
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PROGRAMME
OF MEETINGS ON THE MODULE
One way to explore the contents of 'The Language of Functions and Graphs' is to
arrange a series of departmental meetings. A possible programme. is outlined below.
Meeting 1
What's in the Box?
• Identify the contents of the box and browse through it.
• Consider which classes will use the materials first and arrange that, if possible,
two or more colleagues tryout worksheets Al and A2 (pages 64 and 74 of the
main module book) so that their experiences may be discussed at the next
meeting.
• Arrange
Meeting 2
• Compare
for everyone to have access to the materials over the next few days.
Looking at the Video (issues 1 and 2)
notes and experiences with Worksheet Al and Worksheet
A2.
• Having used Worksheet A2 with classes it will be of interest to see the beginning
of the video tape. This commences with two teachers and their classes working
with Worksheet A2 followed by discussion. Join in the discussion at pauses 1 and
2 on the tape.
• Plan to use further
colleagues.
Meeting 3
• Compare
materials,
including
Worksheet
AS, in parallel
with
Looking at the Video (issues 3 and 4)
classroom experiences,
including sessions using Worksheet AS.
• View the rest of the video tape which shows different approaches to Worksheet
AS and further discussion. Join in discussion pauses 3 and 4.
• Plan some further parallel classroom explorations.
Meeting 4
• Compare
• Explore
Chapter
How Can the Micro Help?
classroom experiences.
the microcomputer
programs using the supporting
4 (this could well take two lunchtime periods).
• Plan some further parallel classroom explorations,
Meeting 5
• Compare
booklets,
and
using the micro if possible.
Tackling a Problem in a Group
classroom experiences.
• The activity on page 207 of the main module book suggests a problem to tackle
together with colleagues. If possible tape record some of the group discussion to
analyse in the next meeting.
• Plan some further parallel classroom explorations,
81
using groupwork if possible.
Meeting 6
• Compare
Ways of Working in the Classroom
classroom experiences.
• Consider Chapter 3 of the Support Materials on page 218 of the main module
book. If you have recorded group discussion from MeetingS, select 3-5 mins. of it
to analyse using the schemes on page 221.
• Discuss ways of managing classroom discussion. Refer to the checklist on the
inside back cover of the main module book.
• Plan some further parallel
discussion, if possible.
Meeting 7
• Compare
classroom
Assessing the Examination
experiences,
including
whole
class
Questions
classroom experiences.
• Chapter 5 of the Support Materials page 234 of the main module book offers a set
of activities to clarify the assessment objectives of the materials and gives
children's scripts for a "marking' exercise. These scripts are also provided in the
pack of "Masters for Photocopying'.
• Plan further activities and meetings.
82
Script A
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Scri pt B
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©Shell Centre for Mathematical
Education, University of Nottingham,
103
1985.
Teaching Strategic
Skills - Publications
List
Problems with Patterns and Numbers - the "blue box" materials
•
School Pack - Problems with Patterns and Numbers 165 page teachers'
book and a pack of 60 photocopying masters.
•
Software Pack - Teaching software and accompanying teaching notes. The
disc includes SNOOK, PIRATES, the SMILE programs CIRCLE, ROSE
and TADPOLES, and four new programs* KA YLES, SWAP, LASER and
FIRST. Available for BBC B & 128, Nimbus, Archimedes and Apple IT
(* the Apple disc only includes the five original programs).
•
Video Pack - A VHS videotape with notes.
The Language of Functions and Graphs - the "red box" materials
•
School Pack - The Language of Functions and Graphs 240 page teachers'
book, a pack of 100 photocopying masters and an additional booklet Traffic:
An Approach to Distance-Time Graphs.
•
Software Pack - Teaching software and accompanying teaching notes. The
disc includes TRAFFIC, BOTTLES, SUNFLOWER and BRIDGES.
Available for BBC B & 128, Nimbus, Archimedes and PC.
•
Video Pack - A VHS videotape with notes.
For current prices and further information please write to: Publications
Department, Shell Centre for Mathematical Education, University of Nottingham,
Nottingham NG7 2RD, England.