masters for photocopying - shell centre for mathematical … · 2010-06-28 · masters for...

The Language of Functions and Graphs

Masters for PhotocopyingCONTENTS

Examination QuestionsThe 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)

Classroom MaterialsUnit A t Al Interpreting points 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)Supplementary booklets:

Interpreting points 24 (100)Sketching graphs from words 26 (102)Sketching graphs from pictures 28 (104)

Unit B 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)Supplementary booklets:

Finding functions in situations 41 (131)Finding functions in tables of data 43 (138)

A Problem CollectionProblems:

Designing a water tank 45 (146)

1

A Problem Collection (cont'd)

The point of no return 47 (150)'Warmsnug' double glazing 49 (154)Producing a magazine 51 (158)The Ffestiniog railway 53 (164)Carbon dating 58 (170)Designing a can 60 (174)Manufacturing a computer 62 (178)The missing planet 64 (182)

Graphs and other data for interpretation: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)

Support MaterialA suggested programme of meetings 816 unmarked scripts for'The hurdles race' 83 (237)Marking record form 86 (236)

"Traffic"-An Approach to Distance-TimeGraphst T1 Taking photographs from a helicopter 88

T2 From photographs to cine film 90T3, T4 More traffic problems 92T5, T6, T7 Acceleration and deceleration 96

"Snapshot.blanks" master 102

Distance-time 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 back to back and folding in half.

©Shell Centre for Mathematical Education, University of Nottingham, 1985.

Note: We welcome the duplication of the materials in this package for useexclusively within the purchasing school or other institution.

2

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THE JOURNEYThe map and the graph below describe a car journey from Nottingham toCrawley using the Ml and M23 motorways.

32

Time (hours)

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Distance(miles)

NOITINGHAM

(i) Describe each stage of the journey, making use of the graph and the map. Inparticular 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 ofthe car varies during the journey.

80

60

Speed(mph)

40

20

o 2 3Time (hours)


4 (12)

CAMPING

On their arrival at a campsite, a group of campers are given a piece of string 50metres long and four, flag poles with which they have to mark out a rectangularboundary for their tent.

They decide to pitch their tent next to a river as shown below ..This means thatthe 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 thelength of the boundary be?

(ii) Describe in words, as fully as possible, how the length of the boundarychanges as the width increases through all possible values. (Consider bothsmall and large values of the width.)

(iii) Find the area enclosed by the boundary for a width of 20 metres and forsome other different widths.

(iv) Draw a sketch graph to show how the area enclosed changes as the widthof the boundary increases through all possible values. (Consider bothsmall and large values of the width.)

AreaEnclosed

Width of the boundary

The campers are interested in finding out what the length and the width of theboundary should be to obtain the greatest possible area.

(v) Describe, in words, a method by which you could find this length andwidth.

(vi) Use the method you have described in part (v) to find this length andwidth.


5 (20)

oI

•Jane

2I

Scale (miles)

•Paul

3I

GOING TO SCHOOL

Graham

•Peter

Jane, Graham, Susan, Paul and Peter all travel to school along the same countryroad 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 aboveshows where each person lives.

The following graph describes each pupil's journey to school last Monday.

6 • •Length 4 •ofjourneyto school(miles) 2 • •

o1

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, 1985.6 (28)

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 isgreater demand.

* the machine is filled up just before the lunch break. (It takes about 10minutes to fill).

Sketch a graph to show how the number of drinks in the machine might varyfrom 8 am to 6 pm.

Number ofdrinksin the machine

I I I , I I I , 1 I • I 11 I

I

8 9 10 11 12 1 2 3 4 5 6

k mornIng )/( afternoon )/

Time of day·


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 belowto show how the car's speed varies along the route.

Peter ~s journey to School

30

Car'sSpeed 20(mph)

10-

I

oI

1 2 4I

5i )

6

I

Di~tance from Peter'~ home (milt:s)


7 (29)

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Distance(metres)

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THE HURDLES RACE

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60Time (seconds)

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The rough sketch graph shown above describes what happens when 3 athletesA, 8 and C enter a 400 metres hurdles race.

Imagine that you are the race commentator. Describe what is happening ascarefully as you can. You do not need to measure anything accurately.


9 (42)

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 iswound from the left hand spool on to the right hand spool.

At the beginning, the radius of the tape on the left hand spool is 2.5 cm. The tapelasts 45 minutes.

(i) Sketch a graph to show how the length of the tape on the left hand spoolchanges with time.

Length of tapeon left handspool

o 10 20 30 40 50

Time (minutes)

(continued)


10 (46)

THE CASSETTE TAPE (continued)

(ii) Sketch a graph to show how the radius of the tape on the left hand spoolchanges with time.

lro.

3 -

Radius of tapeon left handspool (cm)

2 -

1 -

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20I

30I

40"-I /'

50

Time (minutes)

(iii) Describe and explain how the radius of the tape on the right-hand spoolchanges with time.


11 (47)

FILLING A SWIMMING POOL

(i) A rectangular swimming pool is being filled using a hosepipe whichdelivers water at a constant rate. A cross section of the pool is shownbelow.

-_-_-_-_-_- t-----d

~

Describe fully, in words, how the depth (d) of water in the deep end of thepool 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 thepool 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 waterin the pool (d)(metres) 1

o 10 20 30Time (minutes)


12 (52)

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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 sketchingquick 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 'distanceto the horizon'? Does it increase or decrease?

Balloon's height (m) 5 10 20 30 40 50 100 500 1000Distance to horizon (km) 8 11 16 20 23 25 36 80 112

10 20 30 40

Balloon's height

Does this distanceincrease by equal amounts? ...

... or increase by greaterand greater amounts? ..

10 20 30 40

Balloon's height

... or increase by smaller andsmaller 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?

10 20 30 40

Balloon's height

©Shell Centre for Mathematical Education, University of Nottingham, 1985.34 (Ill)

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DESIGNING A WATER TANK

2 metres

2 metres :>

//

A square metal sheet (2 metres by 2 metres) is to be made into an open-toppedwater tank by cutting squares from the four corners of the sheet, and bending thefour remaining rectangular pieces up, to form the sides of the tank. These edgeswill then be welded together.

*

*

How will the final volume of the tank depend upon the size of the squares cutfrom the corners?

Describe your answer by:

a) Sketching a rough graph

b) Explaining the shape of your graph in words

c) Trying to find an algebraic formula

How large should the four corners be cut, so that the resulting volume of thetank is as large as possible?


*

*

DESIGNING A WATERTANK ... SOME HINTS

Imagine cutting very small squares from the corners of the metal sheet. Inyour 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 wordsunderneath:

Volume ofthetank (m3

)

,/

Length of the sides of the squares (m).

*

*

*

In order to find a formula, imagine cutting a square x metres by x rnetresfrom 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, and1 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.


THE POINT OF NO RETURN

Imagine that you are the pilot of the light aircraft in the picture, which is capableof cruising at a steady speed of 300 km/h in still air. You have enough fuel onboard to last four hours.

You take off from the airfield and, on the outward journey, are helped along bya 50 km/h wind which increases your cruising speed relative to the ground to 350km/h.

Suddenly you realise that on your return journey you will be tlying into the windand will therefore slow down to 250 km/h.

What is the maximum distance that you can travel from the airfield, and stillbe sure that you have enough fuel left to make a safe return journey?

:;: Investigate these 'points of no return' for different wind speeds.


*

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

-- 700rJl<l)l-.•...<l)

E 600..Q~-...- 500:g

<l)t;::l-. 400.~E0 300l-.

4-<<l)ut: 200ro•...rJ'J

Cl 100

o 2Time (hours)

3 4

*

Use your graph to find the maximum distance you can travel from theairfield, and the time at which you should turn round.

On the same graph, investigate the 'points of no return' for different windspeeds. 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 speed of the aircraft,one involving wand one involving d and t.

Write down two expressions for the homeward speed of the aircraft,one involving wand one involving d and t.

Try to express d in terms of only t, by eliminating w from the two resultingequations.

Does this explain the pattern made by your 'points of no return '?


"W ARMSNUG DOUBLE GLAZING"

(The windows on this sheet are all Adrawn to scale: 1 em represents 1 foot).

How have "Warmsnug" arrived atthe prices shown on these windows?

Which window has been givenan incorr~ct price? How muchshould it cost?

Explain your reasoning clearly.

£66

r J£46

£55

~_________ £84

~...... I----~J


*

*

*

*

*

*

*

"W ARMSNUG" DOUBLE GLAZING ... SOME HINTS

Write down a list of factors which may affect the price that "Warmsnug" askfor 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 indetermining the price?

Draw scattergraphs to test your ideas. For example, if you think that theperimeter is the most important factor, you could draw a graph showing:

Costofwindow

"-"

Perimeter of window

Does your graph confirm your ideas? If not, you may have to look at someother 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 predictthe price of any window from its dimensions.


PRODUCING A MAGAZINE

A group of bored, penniless teenagers want to make some money by producingand selling their own home-made magazine. A sympathetic teacher offers tosupply duplicating facilities and paper free of charge, at least for the first fewIssues.

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?

How many writers will be needed?

How long will it take to write?

(I pages)

(w writers)

(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 longerit will take to write.

For a fixed length of magazine,the more writers there are,

Complete the statement, and sketch a graphto illustrate it.

Write down other relationships you can find,and sketch graphs in each case.

I~

"-w writers '"

2 The group eventually decides to find out how many potential customersthere are within the school, by producing a sample magazine and conductinga survey of 100 pupils, asking them "Up to how much would you be preparedto pay for this magazine?" Their data is shown below:

Selling price (5 pence) Nothing 10 20 30 40

Number prepared to pay this price (n people) 100 82 58 40 18

How much should they sell the magazine for in order to maximise theirprofit?

3 After a f~w issues, the teacher decides that he will have to charge the pupilslOp per magazine for paper and duplicating.

How much should they sell the magazineJor now?


PRODUCING A MAGAZINE ... SOME HINTS

1 Here is a more complete list of the important factors that must be taken intoaccount:

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 sellingprice at? (s pence)How much profit will we makealtogether? (p pence)How much should we spend onadvertising? (a pence)

2

**

*

***

*

*

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 given in the table of data.

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 theprofit (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.

*

*

3 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 heraised by selling the magazine, (the 'revenue')? How much will thesemagazines cost to produce? How much actual profit will therefore bemade?

Draw up a table of data which shows how the revenue, production costsand 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 pricefor the magazine.


THE FFESTINIOG RAILWAY

This railway line is one of the most famous in Wales.Your task will be to devise a workable timetable forrunning 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 between them are shown in miles)BlaenauFfestiniog

2

Tan-y-Bwlch Z ..P/~

~

*

*

*

Three steam trains are to operate a shuttle service. This means that they willtravel back and forth along the line from Porthmadog to Blaenau Ffestiniog witha IO-minute stop at each end. (This should provide enough time for drivers tochange 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 atspecially designed passing places. (You will need to say where these will beneeded. 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 (includirigstops at intermediate stations. These stops ar~ very short and may be neglected inthe timetabling).

The first train of the day will leave Porthmadog at 9.00 a.m.

The last train must return to Porthmadog by 5.00 p.m. (These times are morerestricted than .those that do, in fact, operate.)


THE "FFESTINIOG RAILWAY" ... SOME HINTS

Use a copy of -the graph paper provided to draw a distance-time graph for the9.00 a.m. train leaving Porthmadog.

Try to show, accurately:

• The outward journey from Porthmadog to Blaenau Ffestiniog.• The waiting time at Blaenau Ffestiniog.• The return journey from Blaenau Ffestiniog to Porthmadog.• The waiting time at Porthmadog ... and so on.

What is the interval between departure times from Porthmadog for the abovetrain '?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 09.00

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

Ask your teacher for a copy of the real timetable, and write about how itcompares with your own.


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56 (167)

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handslgnal to the driver.All trains other than the0840 ex. Porthmadog willalso call on request atBoston Lodge, Plas andDduallt.

Every effort possible willbe made to ensurerunning as timetable butthe Ffestinlog Railway willnot guarantee advertisedconnections nor theadvertised traction in theevent of breakdown orother obslruclion ofservices.

1520 1615 174t1526 1621 175~1550 1640 181~1606 1659 183(1611 1706 183!

FATHER CHRISTMASEXCURSIONS

NOTES ON FFESTINIOGRAILWAY SERVICES

- until 12 May- from 13 May- Saturdays 18 May to

28 September, retimed tePwllheli dep. 0838.Mlnltordd arr. 0911

FO - Fridays only50 - Saturdays onlyBritish Rail services may besubject to alteration at BankHoliday periods.

• 1409 1504 16441415 1510 165(14)5 1530 171(1458 1548 17211507 1600 174(

Special parties and private charterby arrangement.

Please apply to:FFESTINIOG RAILWAYHARBOUR STATIONPORTH MADOGGWYNEDD

Telephone:PORTHMADOG(0766) 2340/2384

I) denotes DIESEL HAULED• I TRAINS - 'ECONOMY'II FARES AVAILABLE.

: Considerable reductionsI for passengers commen-

cing a FULL RETURNJOURNEY by such trains.The return journey maybe made by any train.All other services arenormally steam hauled.

RQ Stops on request only.Passengers wishing toalight should Inlorm theguard before bOlrdlng.Passengers wishing toJoin should give a clear

A special service will operate on21 and 22 December. Details

, available from 1 Oclober.Father Christmas will meet thetrains and distribute presents tothe children.All seats reservable - AdvanceBooking Essential.

1710175118341901

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--~------~~~

09381008

DAILYMonday 16 Sept.

MONDAYS TO FRIDAYS SATS. & SUNS. to Sunday 3 Nov.

1125 1~400 1455 1635:0950 1220 _145509501220

Saturday 31 August to Sunday 15 September

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• 1209 1304 1354 1444 1539 1629 1719 1805 1848

d. 1105d. 1111d. 1133d. 1149d. 1154

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MONDAYS TO THURSDAYS-_._--,--_._---_._~-----------~- ---- -------~-

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it' /,::/41- ~/ / -'mn"l!tttGpti@i f'P+;l@IlQt!Gpt-lffl. t~ "- ~\ \ +~'H''''" 'tod"",:'~' \\ - }' A470Tri •...,t,n,rJrJ\Oolqell"

by Narrow Gauge Train through the Mountains of Snowdonia

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©Shell Centre for Mathematical Education, University of Nottingham, 1985.57 (16S)

CARBON DATING

Carbon dating is a technique for discoveringthe age of an ancient object, (such as a boneor a piece of furniture) by measuring theamount of Carbon 14 that it contains.

While plants and 'animals are alive, theirCarbon 14 content remains constant, but whenthey die it decreases to radioactive decay.

The amount, a, of Carbon 14 in an objectt thousand years after it dies is given bythe formula:

a = 15.3 x- 0.886 t

(The quantity ""a" measures the rate of Carbon 14 atom disintegrations and thisis 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 treeand the other was taken from a charcoal sample found at Stonehenge and is4000 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 behalved?These two answers should be the same, (Why?) and this is called the half-lifeof 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 thepaintings found in the cave.

3 Bones A and B are x and y thousand years old respectively. Bone A containsthree times as much Carbon 14 as bone B.

What can you say about x and y?


CARBON DATING ... SOME HINTS

Using a calculator, draw a table of values and plot a graph to show how theamount of Carbon 14 in an object varies with time.

t (1000's of years) 0 1 2 3 4 5 6 7 8 9 10 ... 17

a (c.p.m)

15

~II---0.0--80..u"-'.•... 10u(]).-'.n0.5~.....-1t::0.nJ-;roU

5~0.•...t::~0E-<

o 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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)

DESIGNING A CAN

A cylindrical can, able to contain half a litre of drink, is to be manufactured fromaluminium. The volume of the can must therefore be 500 cm3•

*

*

Find·the radius and height of the can which will use the least aluminium, andtherefore be the cheapest to manufacture. (i.e., find out how to minimise thesurface area of the can).

State clearly any assumptions you make.

What shape is your can? Do you know of any cans that are made with thisshape? Can you think of any practical reasons why more cans are not thisshape?


60 (174)

*

*

*

*

*

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 tobe 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!).

Using 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 noticeabout your answers? What shape is the can?


MANUFACTURING A COMPUTER

Imagine that you are running a small business which assembles and sells twokinds of computer: Model A and Model B (the cheaper version). You are onlyable 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 atyour company:

Job Title Numper of people Job description Pay Hoursdoing this job worked

Assembler 100 This job involves £100 36 hoursputting the computers per week per weektogether

Inspector 4 This job involves £120 35 hourstesting and per week per weekcorrecting anyfaults in thecomputers beforethey are sold

The next table shows all the relevant data concerning the manufacture of thecomputers.

Model A ModelB

Total assembly time in man-hours 12 6for each computer

Total inspection and correction time 10 30in man-minutes for each computer

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 200of Model B each week.

**

*

What profit are you making at the moment?

How many of each computer should you make in order to improve thisworrying situation?

Would it help if you were to make some employees redundant?


MANUF ACTURING A COMPUTER ... SOME HINTS

1 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?

2 N ow suppose that you manufacture x Model A and y Model B computerseach week.

* 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 ofModel Bcomputersmanufactured(y)

a 100 200 300 400Number of Model A computers manufactured (x)

3 Work out an expression which tells you the profit made on x Model A andy Model B computers.

4 Which points on your graph maximise your profit?


THE MISSING PLANET 1.

In our solar system, there are nine major planets, and many other smaller bodiessuch as comets and meteorites. The five planets nearest to the sun are shown inthe diagram below.

Mars

.. .- .• ~. 'Ii

: ~'::·).~i"... ::;:,,~".Asteroidsf ••• ~~':.' ••.. .. "~'. .

• • •••• #

:- .. .·:.: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 manyyears ago. We shall call this, planet 'X'. In these worksheets, you will try todiscover everything you can about planet 'X' by looking at patterns which occurin 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 thatof our Earth. (So, for example, Saturn is 10times as far away from the Sun as theEarth. Scientists usually write this as 10 A.U. or 10' Astronomical Units').

*

*

*

*

Can you spot any pattern inthe sequence of approximaterelative distances.Can you use this pattern topredict the missing figures?So how far away do you thinkplanet 'X' was from the Sun?(The Earth is 93 million milesaway)Check your completed tablewith the planetary datasheet.Where does the pattern seemto break down?

Planet Relative Distance from Sun, approx(exact figures are shown in brackets)

Mercury ?Venus 0.7 (0.72)Earth 1 (1)Mars 1.6 (1.52)Planet X ?Jupiter 5.2 (5.20)Saturn 10 (9.54)Uranus 19.6 (19.18)Neptune ?Pluto ?


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THE MISSING PLANET ... SOME BACKGROUND INFORMATION

In 1772, when planetary distances were still only known in relative terms, aGerman astronomer named David Titius discovered the same pattern as the oneyou have been looking at. This 'law' was publisheQ by Johann Bode in 1778 andis now commonly known as "Bode's Law". Bode used the pattern, as you havedone, to predict the existence of a planet 2.8 AU from the sun. (2.8 times as faraway from the Sun as the Earth) and towards the end of the eighteenth centuryscientists began to search systematically for it. This search was fruitless untilNew Year's Day 1801, when the Italian astronomer Guiseppe Piazzi discovereda very small asteroid which he named Ceres at a distance 2.76 AU from theSun-astonishingly close to that predicted by Bode's Law. (Since that time,thousands of other small asteroids have been discovered, at distances between2.2 and 3.2 AU from the sun.)

In 1781, Bode's Law was again apparently confirmed, when William Herscheldiscovered the planet Uranus, orbitting the sun at a distance of 19.2 AU, againstartlingly 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 distantplanets.

However, when Neptune and Pluto were finally discovered, at 30 AU and 39A 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, 1985.66 (1~4)


Look at the Planetary data sheet, which contains 7 statistics for each planet.

The following scientists are making hypotheses about the relationship betweenthese statistics:

AThe further a

B C The smallerplanet is away from the planet

@ the Sun, the longer G the slower iteit takes to orbit ct ••• J spIns.the Sun.

Do you agree with these hypotheses? How true are they? (Use the datasheet)

* 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 someidea of how strong the relationship is between the two variables.

For example, here is a 'sketch' scattergraph testing the hypothesis of scientist A:

.~.n1-0oo.•...

~

.X 1-0.•••• Cl)Cl) .••••c:: .-

m ro 0..1-0,_ ::sro ~ ~

~X ? XDistance from sun

C)c::::s.•...0.Cl)

ZX

xo.•...::s0: Notice that:

There does appear to be arelationship between the distancea planet is from the Sun and thetime it takes to orbit once. Thehypothesis seems to be confirmed.

We can therefore predict the orbitaltime for Planet X. It should liebetween that of Mars (2 years) andJupiter (12 years). (A more accuratestatement would need a more accurategraph. )

• Sketch scattergraphs to test your own hypotheses. What else can be foundout about Planet X? What cannot be found?



After many years of observation the famousmathematician Johann Kepler (1571-1630)found that the time taken for a planet to orbitthe Sun (T years) and its average distancefrom the Sun (R'miles) are related by theformula

where K is a constant value.

planet

*

*

*

Use a calculator to check this formula from the data sheet, and find the valueofK.

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 thesesheets).

We asserted that the orbits of planets are 'nearly circular'. Assuming this isso, 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)?

(Hint: Find out how far the planetmoves during one orbit. You canwrite this down in two differentways using R, T and V)(Warning: Tis in years, Vis inmiles per hfJur)

Use a calculator to check your formula from the data sheet.Use your formula, together with what you already know about Rand T, tofind a more accurate estimate for the speed of Planet X.

Assuming that the planets are spherical, can you find a relationshipconnecting

- 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.


FEELINGS

Her timetable for the day was as follows:

These graphs show how a girl's feelings variedduring a typical day.

7 .()() amX.()() aml) .()() aml).3() am

1O.3() amI I.()() am12 .()() am

I .3() pm2ASpm

.3.()() pm.+.()() pmh. ()()pm7 .()() pm

I ().3() pm

woke upwent to schoolAssemblyScienceBreakMathsLunchtimeGamesBreakFrenchwent homedid homeworkwent lO-pin bowlingwent to bed

Happy

'"0oo

~

(a) Try to explain the shapeof each graph, as fully aspossible.

Sad (b) How many meals did she eat?I I I I I I I I I I I I I I I I I I Which meal was the biggest?6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11

Time of day Did she eat at breaktimes?

Full ofHow long did she spend

energy eating lunch?Which lesson did she enjoy

;>, the most?OJ)~When was she ""tired andC)

c:U.J depressed?" Why was this?

Exhau~teJ When was she ""hungry butI I I I I I I I I I I I I I I , I I happy?" Why was this?6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 1011

Time of dayMake up some more questions

Full like these, and give them toup your neighbour to solve.

~C)OJ) f (c) Sketch graphs to show howc:::l

:r: your feelings change during

Starvingthe day. See if your

, I I I I I I I t I 1 I I I I I Ineighbour can interpret them

6 7 8 <) 1() 11 12 I :2 3 4 6 7 8 <) 1 () 11 correctly.Time of day

©Shell Centre for Mathematical Education, University of Nottingham, 1Y~5.

6Y ( 1Y I)

THE TRAFFIC SURVEY

A survey was conducted to discover thevolume of traffic using a particularroad. The results were published inthe form of the graph which shows thenumber of cars using the road at anyspecified time during a typicalSunday 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 youknow of, which may produce such a graph.)

900

800"0

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I~ Sunday -- ~ Monday ~


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THE MOTORW AY JOURNEY

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" f' '-....•..•...,.....•.•...

~.....•.•...

~.....•.•...

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a 50 100 150 200 250Distance travelled in miles

The above graph shows how the amount of petrol in my car varied during amotorway journey.

Write a paragraph to explain the shape of the graph. In particular answer thefollowing 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 andthen 10 miles through a city, where I had to keep stopping and starting. Alongcountry 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.


GROWTH CURVES

Paul and Susan are two fairly typical people. The following graphs compare howtheir weights have changed during their first twenty years.

I I I I I/

~Paul /Susan----

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10

o 5 10

Age in years

15 20

Write a paragraph comparing the shape of the two graphs. Write downeverything you think is important.

Now answer the following:

1 How much weight did each person put on during their "secondary school"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?

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, 1985.72 ( 19-4)

ROAD ACCIDENT STATISTICS

The following four graphs show how the number of road accident casualties perhour varies during a typical week.

Graph A shows the normal pattern for Monday, Tuesday, Wednesday andThursday.* Which graphs correspond to Friday, Saturday and Sunday?* Explain the reasons for the shape of each graph, as fully as possible.* What e.vidence IS there to show that alcohol IS a major cause of road

accidents?

Graph AMonday to Thursday

Graph B120

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73 (195)

THE HARBOUR TIDE

The graph overleaf, shows how the depth of water in a harbour varies on aparticular 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 varyfrom the average?

2 Ships can only enter the harbour when the water is deep enough. Whatfactors 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 withcargo and only 2 metres when unloaded.

Discuss when it can safely enter and leave the harbour.

Draught·····::· ...-::~·::.:::.:··.·.·::.' .

Harbour depth

Make a table showing when boats of different draughts can safely enter andleave the harbour on Wednesday.

3 Try to complete the graph in order to predict how the tide will vary onThursday. How will the table you draw up in question 2 need to be adjustedfor Thursday? Friday? ...

4 Assuming that the formula which fits this graph is of the formd = A + B cos(28t + 166t

(Where d = depth of water in metrest = 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?


74 (196)

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ALCOHOL

Read through the data sheet carefully, and then try to answer the followingquestions:

U sing the chart and diagram on page 2, describe and compare theeffects 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 ofbeer. He takes a taxi home and goes to bed. Can he legally drive towork at 7 am the next morning? When would you advise him that he isfit to drive? Explain your reasoning as carefully as possible.

The five questions below will help you to compare and contrast the informationpresented on the data sheet.

1 Using only the information presented in words by the "Which?" report,draw an accurate graph showing the effect of drinking 5 pints of beerat 2 am.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.

2 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 your answers with those already obtained.

3 Using only the table of data from the AA bC!0kof driving,draw another graph to show the effect of an 11stone man drinking 5 pints ofbeer.

ComP':l~e this graph to those already obtained.

Answer 1a) b) c) d) from this graph, and compare your answers with thoseabove.


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ALCOHOL (continued)

4 Compare the graph taken from the Medical textbook with those drawn forquestions 1,2 and 3. Answer question la) b) c) and d) concerning the 11stone 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 (does it take up much room?)Accuracy (is the information over-simplified?)Simplicity (is it easy to understand?)Versatility (can it show the effects of drinking different

amounts of alcohol easily?)Reliability (which set of data do you trust the most?

Why? Which set do you trust the least? Why?)

A business woman drinks a glass of sherry, two glasses of table wineand a double brandy during her lunch hour, from 1pm to 2 pm. Threehours later, she leaves work and joins some friends for a meal, whereshe drinks two double whiskies.

Draw a graph to show how her blood/alcohol level varied during theentire afternoon (from noon to midnight). When would you haveadvised her that she was unfit to drive?


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A SUGGESTED PROGRAMME OF MEETINGS ON THE MODULE

One way to explore the contents of 'The Language of Functions and Graphs' is toarrange 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 themain module book) so that their experiences may be discussed at the nextmeeting.

• Arrange for everyone to have access to the materials over the next few days.

Meeting 2 Looking at the Video (issues 1 and 2)

• Compare notes and experiences with Worksheet Al and Worksheet A2.

• Having used Worksheet A2 with classes it will be of interest to see the beginningof the video tape. This commences with two teachers and their classes workingwith Worksheet A2 followed by discussion. Join in the discussion at pauses 1 and2 on the tape.

• Plan to use further materials, including Worksheet AS, in parallel withcolleagues.

Meeting 3 Looking at the Video (issues 3 and 4)

• Compare classroom experiences, including sessions using Worksheet AS.

• View the rest of the video tape which shows different approaches to WorksheetAS and further discussion. Join in discussion pauses 3 and 4.

• Plan some further parallel classroom explorations.

Meeting 4 How Can the Micro Help?

• Compare classroom experiences.

• Explore the microcomputer programs using the supporting booklets, andChapter 4 (this could well take two lunchtime periods).

• Plan some further parallel classroom explorations, using the micro if possible.

Meeting 5 Tackling a Problem in a Group


• The activity on page 207 of the main module book suggests a problem to tackletogether with colleagues. If possible tape record some of the group discussion toanalyse in the next meeting.

• Plan some further parallel classroom explorations, using groupwork if possible.

81

Meeting 6 Ways of Working in the Classroom


• Consider Chapter 3 of the Support Materials on page 218 of the main modulebook. If you have recorded group discussion from MeetingS, select 3-5 mins. of itto analyse using the schemes on page 221.

• Discuss ways of managing classroom discussion. Refer to the checklist on theinside back cover of the main module book.

• Plan some further parallel classroom experiences, including whole classdiscussion, if possible.

Meeting 7 Assessing the Examination Questions


• Chapter 5 of the Support Materials page 234 of the main module book offers a setof activities to clarify the assessment objectives of the materials and giveschildren's scripts for a "marking' exercise. These scripts are also provided in thepack of "Masters for Photocopying'.

• Plan further activities and meetings.

82

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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. Thedisc includes SNOOK, PIRATES, the SMILE programs CIRCLE, ROSEand TADPOLES, and four new programs* KAYLES, SWAP, LASER andFIRST. 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. Thedisc 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: PublicationsDepartment, Shell Centre for Mathematical Education, University of Nottingham,Nottingham NG7 2RD, England.

masters for photocopying - shell centre for mathematical … · 2010-06-28 · masters for...

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