lfg interpreting sketching

17
4. Sport Al INTERPRETING POINTS Suppose you were to choose, at random. 1(X) people and measure how heavy they are. You then ask them to perform in 3 sports; High Jumping. Weight Lifting and Darts. Sketch scattergraphs to show how you would expect the results to appear. and explain each graph. underneath. Clearly state any assumptions you make. Max Score with 3 darts Body weight These four shapes each have an area of 36 square untis. ~ Label four points on the graph below. with the letters A. B. C and D. Can you draw a fifth shape. with an area ot 36 square units, to correspond to the other point? Explain. Draw a scattergraph to sho~ nery reciangle with an area of 36 square units. height * Finally, what happens if you include all shapes. with the same area, on your graph? As you work through this booklet, discuss your answers w iih sour neighbours and try to come to some agreement. I. The Bus Stop Queue Who is represented by each point on the scatitrgrapli. below! Max Height Jumped Max Weight Lifted Body weight C. Alice Brenda Cath~ 1)cniiis I izol I icd;i (in iii /.‘ I. 2• Age 4. 5. 6. 7. . . . v~idtli Height lire for Mathematical Education. University of NottinL!ham 985

Upload: eemina13

Post on 10-Apr-2015

278 views

Category:

Documents


4 download

TRANSCRIPT

Page 1: Lfg Interpreting Sketching

4. SportAl INTERPRETING POINTS

Suppose you were to choose, at random. 1(X) people and measurehow heavy they are. You then ask them to perform in 3 sports;

High Jumping. Weight Lifting and Darts.

Sketch scattergraphs to show how you would expect the results toappear. and explain each graph. underneath. Clearly state anyassumptions you make.

MaxScorewith3 darts

Body weight

These four shapes each have an area of 36 square untis.~ Label four points on the graph below. with the letters

A. B. C and D.

Can you draw a fifth shape. with an area ot 36 square units, tocorrespond to the other point? Explain.

Draw a scattergraph to sho~nery reciangle with an area of 36square units. height

* Finally, what happens if you

include all shapes. with the samearea, on your graph?

As you work through this booklet, discuss your answers w iih sourneighbours and try to come to some agreement.

I. The Bus Stop Queue

Who is represented by each point on the scatitrgrapli. below!

MaxHeightJumped

MaxWeightLifted

Body weight

C.

Alice Brenda Cath~ 1)cniiis I izol I icd;i (in iii

/.‘

I. 2•

Age

4. 5.6.

7.

. .

.

v~idtliHeight —

lire for Mathematical Education. University of NottinL!ham 985

Page 2: Lfg Interpreting Sketching

2. Two Aircraft 3. Telephone Calls

The following quick sketch graphs describe two light aircraft, Aand B: (note: the graphs have not been drawn accurately)

Cruising RangeSpeed

•A

Age Size Passenger CapacityThe first graph shows that aircraft B is more expensive thanaircraft A. What else does it say?* Are the following statements true or false?

“The older aircraft is cheaper”?“The faster aircraft is smaller”?“The larger aircraft is older”?“The cheaper aircraft carries fewer passengers”?

* Copy the graphs below. On each graph, mark and label twopoints to represent A and B.

OneFivecallscountry.They recorded both the cost oftheir calls, and the length oftime they were on thetelephone, on the graph below:

> Duration of call

* Who was ringing long-distance? Explain your reasonincarefully.

* Who was making a local call? Again, explain.* Which people were dialling roughly the same distance

Explain.* Copy the graph and mark other points which show peopl

making local calls of different durations.* If you made a similar graph showing every phone call made i

Britain during one particular week-end, what would it loolike? Draw a sketch, and clearly state any assumptions yomake.

3

Cost

weekend,people made telephoneto various part of the

•B

Vi

Cost i ‘~

ofcall

• John • Barbara

Clare

• David • Sanjay

SizeAge

Cruising Speed2

Range

Shell Centre for Malhemaiieal Educalion. Univershy ofNottingham. 985.

Page 3: Lfg Interpreting Sketching

A2 ARE GRAPHS JUST PICTURES?Finally. discuss and write about this problem

Choose the best answer from the following and explain exactlyhow it fits the graph.

Write down reasons why you reject alternatives

FishingPole Vaulting100 metre SprintSky DivingGolfArcheryJavelin ThrowingHigh JumpingHigh DivingSnookerDrag RacingWater Skiing

. \change as it flies through the air ii

* Discuss this situation with your neighbour, and write down ~clear description stating how you both think the speed of thc

*golf ball changes.Now sketch a rough graphto illustrate yourdescription:

Speedoftheball

Time alter the ball ishit by the goliclub.

Which Sport?

Which sport will produce a graph like this?

Speed

Time

Golf Shot

I-low does the speed of the ballthis amazing golf shot?

ThATAPIc.TURI1OF 13IE 1’fl1N1~N ~

L~ttvFELLOFF?~~~k

Shell Centre br Maihemancal Fducabion. University of Nottingham. 19H5

Page 4: Lfg Interpreting Sketching

Peter attempted the golf questionand produced a graph like this:

* Comment on it.

* Can you suggest why Peter

drew his graph like this?

* Can you see any connection between

the cartoon on page 1?

Now try the problem below:

Describe your answer both in words and by sketching agraph in your hook.

Speed of theRoller-coaster

i~-~- I I I I

A B C D E F GDistance travelled along the track

This next activity will help you to see how well you have drawnyour sketch graph.

Fold this booklet so that you cannot see the picture of the roller-coaster track.

Try t~ answer the following questions using only your sketchgraph.

* Along which parts of the track was the roller-coaster travelling

quickly? slowly?

* Was the roller-coaster travelling faster at B or D?

D or F? C or• Where was the roller-coaster accelerating (speeding up)?

decelerating (slowing down)?

Check your answers to these questions by looking back at thepicture of the roller-coaster track. If you find any mistakes.redraw your sketch graph. (It is better to use a fresh diagram thanto try and correct your first attempt.)

* Now invent some roller-coaster tracks of your own.

Sketch a graph for each one, on a separate sheet of paper. Passonly the sketch graphs to your neighbour.Can she reconstruct the shape of the original roller-coastertracks?

* Do you notice any connection between the shape of a roller-

coaster track, and the shape of its graph? If so write down anexplanation.Are there any exceptions?

Speedof ball

Time after ball is hitPeter’s attempt and

A B Roller-coaster

‘—I

-‘I

The picture above shows the track of a roller-coaster, which istravelling at a slow constant speed between A and B. l-Io~ willthe speed of (his roller-coaster van as ii travels along the trackfrom A to 0?

3

Slid t (‘cii rc or M a I hcmai,cal Lducaiion. U 01’ crsii~ of Non ingharn. 1985

Page 5: Lfg Interpreting Sketching

Sketch graphs to illustrate the following statements. Labelyour axes with the variables shown in brackets. For the laststatement you are asked to sketch two graphs on the sameaxes.

“In the spring, my lawn grew very quickly and it needed cuttingevery week, but since we have had this hot dry spell it needscutting less frequently.”

(length of grass/time)

“When doing a jigsaw puzzle, I usually spend the first half anhour or so just sorting out the edge pieces. When I have collectedtogether all the ones I can find, I construct a border around theedge of the table. Then I start to fillin the border with the centrepieces. At first this is very slow going but the more pieces you putin, the less you have to sort through and so the faster you get.”

(number of pieces put in jigsaw/time).

“The Australian cottony cushion scale insect was accidentallyintroduced into America in 1868 and increased in number until itseemed about to destroy the Californian citrus orchards where itlived. Its natural predator, a ladybird, was artificially introducedin 1889 and this quickly reduced the scale insect population.Later, DDT was used to try to cut down the scale insectpopulation still further. However, the net result was to increasetheir numbers as, unfortunately, the ladybird was far moresusceptible to DDT than the scale insect! For the first time in fiftyyears the scale insect again became a serious problem.”Use the same axes.(scale insect population/time); (ladybird population/time).

A3 SKETCHING GRAPHS FROM WORDS

Total time itwill take tofinish thejob

Number of people picking strawberries

* Compare your graph with those drawn by your neighbours.Try to come to some agreement over.a correct version.

* Write down an explanation of how you arrived at your answerIn particular, answer the following three questions.

— should the graph ‘slope upwards’ or ‘slope downwards”Why?

— should the graph be a straight line? Why?

— should the graph meet the axes? If so, where?

1 If not, why not?

00

GOL’J

Picking Strawberries

The more people we get to heLp,the sooner we’Ll finish picking

these strawberries.

r5TRAWgERRY

PICKERSWANTED,,~

-IL-IL

* Using axes like the ones below, sketch a graph to illustrate thissituation.

©ShelI Centre ror Maihema lea (Ion. UniversflyotNottingham. I9~S,

Page 6: Lfg Interpreting Sketching

9. the speed of a girl vary on a swing?10. the speed of a ball vary as it bounces along?

(a) (c)

1. “Prices are now rising more slowly than at any time duringthe last five years.”

2. “I quite enjoy cold milk or hot milk, but I loathe lukewarmmilk!”

3. “The smaller the boxes are, then the more boxes we can loadinto the van.”

4. “After the concert there was a stunned silence. Then oneperson in the audience began to clap. Gradually, thosearound her joined in and soon everyone was applauding and

-~ cheering.”‘~ 5. “If cinema admission charges are too low, then the owners

will lose money. On the other hand, if they are too high thenfew people will attend and again they will lose. A cinemamust therefore charge a moderate price in order to stayprofitable.”

In the following situations, you have to decide whathappens. Explain them carefully in words, and choose thebest graph, as before.

How does...

6. the cost of a bag of potatoes depend on its weight?7. the diameter of a balloon vary as air is slowly released from

it?8. the time for running a race depend upon the length of the

race?2

Choose the best graph to fit each of the ten situationsdescribed below. (Particular graphs may fit more thanone situation.) Copy the graph, label your axes andexplain your choice, stating any assumptions you make.If you cannot find the graph you want, draw your ownversion.

3

h,r \1,jjI,cni.iipc~iI I dut.iiic,n. t npvcr’uI~ NotungIn.ni. 19H5,

Page 7: Lfg Interpreting Sketching

The Big Wheel in thediagram turns roundonce every 20 seconds.On the same pair ofaxes, sketch twographs to show howboth the height of carA and the height of carB will vary during aminute.

Describe how your graphs will change if (he whcel(urns more quickly.

OrbitsEach of the diagrams below shows a spacecraft orbiting aplanet at a constant speed.Sketch two graphs to show how the distance of thespacecraft from the planet will vary with time.

7

$ Using a dotted line on the same\( axes, show how your graphs ~

will change if the speed of the ~spacecraft increases as it getsnearer to the planet.

Now invent your own orbits and sketch their graphs.on a veparate slicer of paper. Give only your graphsto sour neighbour. Can she reconstruct the orbitsirom the graphs alone!

A4 SKETCHING GRAPHS FROM PICTURES

Explain your answer in eachand with a sketch graph.assumptions that you make.

-~ Distance along track

Compare your graphs with those produced byyour neighbours. Try to produce three graphswhich you all agree are correct.

The Big 4% heel

A

Motor Racing

How do you think the speed of a racing carwill vary as it travels on the second laparound each of the three circuits drawnbelow? (S = starting point)

a

~s ~

Circuit I Circuit 2 Circuit 3

case both in wordsState clearly any

I

Speed

Shell (enire br Mathematical Education, fly fNottingham, 1985.

Page 8: Lfg Interpreting Sketching

Look again at the graph you drew for the third circuit. Inorder to discover how good your sketch is, apswer thefollowing questions looking only at your sketch graph.When you have done this, check your answer by lookingback at the picture of the circuit. If you find any mistakesredraw your sketch graph.

— Is the car on the first or second lap?

— How many bends are there on the circuit?

— Which bend is the most dangerous?

— Which “straight” portion of the circuit is the longest?Which is the shortest?

— Does the car begin the third lap with the same speedas it began the second? Should it?

The graph below shows how the speed of a racing car variesduring the second lap of a race.

Speed

Which of these circuits was it going round?

Discuss this problem with your neighbours.Write down your reasons each time you reject a circuit.

3

Distance along the track

Now invent a racing circuit of your own with, at most,four bends.Sketch a graph o.n a separate sheet ofpaper to show howthe speed of a car will vary as it goes around your circuit.Pass only your graph to your neighbour.Can she reconstruct the shape of the original racingcircuit?

srv~cv]

2

Shell Centre for Muihemailcal Education. UniversiiyofNowngham. 19N5

Page 9: Lfg Interpreting Sketching

* Draw sketch graphs for the following sequence of

bottles.

* Using your sketches explain why a bottle with straight slopingsides does not give a straight line graph (ie: explain why the inkbottle does not correspond to graph g).

* Invent your own bottles and sketch their graphs on a separatesheet of paper.

~j Pass only the graphs to your neighbour.Can he reconstruct the shape of the original bottles using only

~ your graphs?If not, try to discover what errors are being made.

* Is it possible to draw two different bottles which give the sameheight-volume graph?Try to draw some examples.

A5 LOOKING AT GRADIENTS

Filling Bottles

In order to callibrate a bottle so that itmay be used to measure liquids, it isnecessary to know how the height ofthe liquid depends upon the volume inthe bottle.

The graph belowshows how the heightof liquid in beaker X varies as water issteadily dripped into it. Copy thegraph, and on the same diagram, showthe height-volume relationship forbeakers A and B.

And two more for E and F...

~Lzz~Volume

LiBeaker X

4-

U

AS Volume

Sketch two more graphs for C and D.

4-

ULi ~LJ x

BeakerX C D Volume

BcakerX E

©SheII Centre or at emacica ucation. niversity o ottingham. 1985.

Page 10: Lfg Interpreting Sketching

4. Sharks and Fish INTERPRETING POINTS

13e low is a simplified description ofwhat can happen when two speciesinteract. The sharks are thepredators and the fish are the prey.The situation in statement A hasbeen represented on the graph by a ____________

point.

H ow does this point move as timegoes by?

D&o,sL Aszs wu.4~~ .4asenah&aFdL

4$ d’a tw,~ ~‘a &n ataa.~~ .a..Jt

a rca.

(C) The sharks eat many of the fish until..

(D) . . .the fish population is insufficient to support all the sharks.Many sharks therefore decide to leave.

(E) With few sharks around, the fish population increases onceagain.

(F) The area now contains enough food to support more sharks, sothey return.

(G) and begin to eat the fish ... until. ______________________

Examination mark4

-4

t3

I. School Reports

1~Iex has ba~ ew6.tnia4 t°3y a.u fon~a.,d H.45 hAS ‘a a-~po~v t4C0a.aamt pf.s-(.o.r.a.n Ce

Number of sharks(predators)

Number of fish (prey)(A) Due to the absence of many sharks, there is an abundant

supply of fish in the area.

(B) Sensing a plentiful supply of fish for food, sharks enter the

o.tit pupiL, as 1w e~J(MoAt darfy shows, 6t4 hjtr C cc.,irata

a.,s bü,a~ous t~ ~

pnr. Witk nia. t~,t, she ~ a~a

ca Ut4 sa$jat. _______

t.ka t,.41. a.-d. dssnva4k.s ,nojviouj ea-,...ata,,. .a,.U

Wt~L Dn -

Each school report is represented by one of the points in the graphbelow. Label four points with the names Alex. Suzy. Catherine andDavid. Make up a school report for the remaining point.

.

Effort .~4 5•

©SheII Centre For Mahematieal Edoch~Ion. University of Nottingham. 1985

Page 11: Lfg Interpreting Sketching

2. Is Height Hereditary? 3. BagS of Sugar

n an experiment. 192 fathers and sons were measured.The sons were measured when they had attained their full adulteight.)What can you say about points A and B? Cost DWhat conclusions can be drawn from this graph? A

. 8.

Weight

72 Each point on this graph represents a bag of sugar.(a) Which bag is the heaviest?(b) Which bag is the cheapest?(c) Which bags are the same weight?(d) Which bags are the same price?(e) Which of F or C would give better value for money?

How can you tell?(f) Which of B or C would give better value for money?

How can you tell?(g) Which two bags would give the same value for money?

How can you tell?

64

62 64 66 68 70 12

Height of father (inches)

2

. .

. .

. . S

e • S

. . .. .

.

. .

S.

• S

•5

S.

.• S

• •

SS

• S

• . ••

55J

• • S. S S

• .S • •

• .

• S.

• •5

• .

• •5S S

• S S

• S SS• S

55 S

S

• S S

• .

• •A. • S•

. .I •

• S S

S S

• . S •

. . S

. •S

• I

I I I

;helI Centre for Mathematical Education. University of Nottingham. 1985.

Page 12: Lfg Interpreting Sketching

SKETCHING (;RAPHS FROM WORI)S

height ol

Sketch graphs to illustrate the following situations.You have to decide on the variables and the relationshipsinvolved. Label your axes carefully, and explain yourgraphs in words underneath.

o~ does

Your height vary with age?

The amount of dough needed to make a pu/a depend upon itsdiameter?

The amount of daylight we get depend upon the time of year?

The number of people in a supermarket ~ary during a typicalSat urday?

The speed of a pole-vaulter vary during a typical jump?

The waler level in your bathtub vary. before, during and alteryou take a bath?

Hoisting the flag

Every morning, on the summer camp, the youngest boy scout hasto hoist a flag to the top of the flagpole.

(i) Explain in words what each of the graphs below would mean.(ii) Which graph shows this situation most realistically? Explain.

(iii) Which graph is the least realistic? Explain.

(a) I-Ic igh IIi a g

0q

C)

0

(b)

4~ini e

Height offlag

Height olflag

Height offlag

I tue

Hag[leight of

(e)

I inie

(f)

I iine1 n~e

hell Centre for Mathematical Education. University ol Nottingham. 985.

Page 13: Lfg Interpreting Sketching

Choose the best graph to describe each of the situationslisted below. Co~y the graph and label (he axes clearly withthe variables shown in brackets. If you cannot lind (he graphyou want, then draw your own version and explain it fully.1

) The weightlifter held the bar over his head for a few unsteadyseconds, and then with a violent crash he dropped it. (height ofbar/time)

~) When I started to learn the guitar, I initially made very rapidprogress. But I have found that the better you get, the moredifficult it is to improve still further. (proficiency/amount ofpractice)

If schoolwork is too easy, you don’t learn anything from doingit. On the other hand, if it is so difficult that you cannotunderstand it, again you don’t learn. That is why it is soimportant to pitch work at the right level of difficulty.(educational value/difficulty of work)

When jogging, I try to stan off slowly, build up to a comfortablespeed and then slow down gradually as I near the end of asession. (distance/time)

) “In general, larger animals live longer than smaller animals andtheir hearts beat slower. With twenty-five million heartbeatsper life as a rule .of thumb, we find that the rat lives for onlythree years, the rabbit seven and the elephant and whale evenlonger. As respiration is coupled with heartbeat—usually onebreath is taken every four heartbeats—the rate of breathing alsodecreases with increasing size. (heart rate/life span)

) As for 5, except the variables are (heart rate/breathing rate)

2

Now make up three stories of your own to accompany threeof the remaining graphs. Pass your stories to yourneighbour. Can they choose the correct graphs to go withthe stories?

3

Shell Cenire for Mathematical Education. University ot Nc,Iaini,haoi 19NS

Page 14: Lfg Interpreting Sketching

SKET(IIIN(; CRAPIISFROM P1(11 iu~:s In theaccompanylnghookla,particlcsaiemovingajonganumhcr

Parlicics and Paths of different paths.Qr

For each situation:* Sketch a rough graph to show how the distance from B will vary

with the distance from A.Qq Qt

I 0

Distancefrom B(cm)

B 5A

n the diagram above, there are 5 particles labelled p. q, r, sand t.

Without measuring, can you label each point on the graph belowwith the correct letter?

0 5 10Now check your answer by measurement Distance from A (cm)(A and Bare 6cm apart)

10 * Check your answer by measuring various positions, recording

your answers in a table and by plotting a few points accurately.~Thsta~ce * Try to find a formula which describes the relationship between

the two distances.(cm)

Continue exploring other paths and their graphs.

Write up all your findings.

0 5 10 4Distance from A (cm)

Shell Centre for Mathematical Educaijon. UrnversicyotNottingham, 1985.

Page 15: Lfg Interpreting Sketching

A B

xIn this diagram. particle xis moving slowly along the path shown bythe dotted line, from left to right.

* Sketch a graph w show how the distance from B relates to (hedistance from A during this motion.

0 10Distance from A (cm)

* Check your answer by measuring variousrecording them in the table:

Try to mark the positions of the five particles a. b, c, d and eon thright hand diagram (b has been done for you).

* Which positions are impossible to mark? Why is this? Try umark other points on the graph which would give impossiblipositions on the diagram. Shade in these forbidden regions oithe graph.

* One position of particle b has been shown. Is this the onl’position which is 4 cm from both A and B? Mark in any othepossible positions for particle b.

* Which points on the graph give only one possible position on th~diagram?

3

Distancefrom B(cm)

Diagram

10

5

a

Graph

e

d

k)‘C

C

EU

~,I0

0I

bc A

5Distance

10from A (cm)

Distance from A(cm)

positions and

E)istance from(cm)

Write down any formulae that you can find which fit yourgraph.

2

Slit_il ( enire fur M;iIlicrn~iipc;il lduc,ilpnn_ L nivcrsii~ ul Niilltngham. urns

Page 16: Lfg Interpreting Sketching

SKETCHING GRAPHS FROM PICTURES (contd)

Particles and Paths(vi)A

(I) .

/

/I

II

• II

S

/S /

SF4. B

—— ——

,GTr *4S

/S

continue exploring other (ii) ~IIand their graphs.

up all your I indings I A

II

I/

S\

4.

,0

F-

5©SheII Centre for Mathematical Education, University of Nottingham, 1985.

Page 17: Lfg Interpreting Sketching

(iii)4- ‘S

6’ ‘5

‘5/ ‘5

/// I’

IA 13

III ‘

/\ /

/,

‘5-

A— —— —

-— —Q~(Iv) — —

S.S.

F •4

II

I

IA B

‘ I/

I/

/S.‘S F

——I

6

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