ZXTape! 0Created with Ramsoft MakeTZXmh4 BBt"B6:6:0: maths help tape 4  G C E Tutoring  Dec 1983 r(10 ) Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ $:"MATHS help mh4"::"STATISTICS" :" G C E Tutoring Jan 1984" 2n:"Hello. What's your name ?":n$:n$:"OK, ";n$;",":"I'll begin by telling you how touse this maths tape" 4T:"This tape covers 7 subjects which all come under the headingof statistics." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AG:"Thanks. From now on when I want you to press the C key, I'll say:" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"pie charts bar charts frequency distribution histograms frequency polygons averages probability tree diagrams" K2:"Which subject do you want to try ?":s$: L%s$>3s$=s$(13) M 96` Ns$="pie"100d Os$="bar"200 Ps$="ave"300, Qs$="pro"500 Rs$="his"800  Ss$="fre"900 Ts$="tre"700 Us$="fre"1000 Z _ 70F `Fj=1̱(s$):(s$(jj))<97as$(jj)=(32 +(s$(jj))) aj: c d f i pie charts l mn$="pie charts" n:"Data of any kind is worthless if it is presented misleadingly.What is needed is a form of presentation that is easily understood."::"To the non-mathematician - or inother words most people - the PIE CHART is the clearest method" pK:"A pie chart is a picture showingthe kind of data in the table below" qscrhdr s!:"Activity hours a week": vr"Sleep 56 Work 37 Leisure 57 Other 18" x502,67C,40( z-40(,0 }-20,38& 20,-38& -35#,-7 35#,7 -24,-24 10 ,7;"s" 12 ,3;"w" 16,7;"l" 15,3;"o" :: scrhdr i"The proportion of each segment of the pie chart represents the share out of the 168 hours in a week." <:"The angle of each segment is calculated like this :"  "total angle = 360 degrees total number of hours = 168 therefore 360/168 degrees/hour = 2.14 angle per segment = hours in segment x degrees per hour" scrhdr *start=100d:n$="pie charts":endsec    bar charts  n$="bar charts" /:"Data of any kind is worthless if it is presented misleadingly.What is needed is a form of presentation that is easily understood."::"To the non-mathematician - or inother words most people - the BAR CHART is a clear method that provides more detailed information than the pie chart" S:"A histogram is a form of graph showing the kind of data in the table below" scrhdr !:"Activity hours a week": r"Sleep 56 Work 37 Leisure 57 Other 18" :" sleep work leis. other " " 0 25 50" scrhdr +start=200:endsec , . 1 averages 4 5n$="averages" 6v:"There are 3 sorts of averages. The most commonly used one is the mean, which most lay-peoplecall the average." 8n:"The MEAN is calculated like this"::"sum of the values /number of values" ;:"eg. the mean weight of 5 pupils who weigh 51kg, 47kg, 63kg, 66kg and 52kg is given by mean =(51+47+63+66+52)/5=53.8 kg" = >qsthdr @["what is the mean of ":r(1);" ";r(2);" ";r(3);" and ";r(4);" ? " B;exp=(r(1)+r(2)+r(3)+r(4))/4 C 9500% Eloop=320@ Freturn=330J Hgetans Jscrhdr L"Another type of average is the MODE. It represents the most popular value, or in other wordsthe one with the highest frequency." Oy:"eg. the mode in a survey showing : dress size %sales" R T" 8 8 10 14 12 26 14 34 16 10 18 8" V"The mode is size 14" Yscrhdr \"The last type of average, and the most complex, is the MEDIAN.The median is the value which comes in the middle of a set of values, when they are placed in ascending order. Don't panic ! This is not as bad as it sounds" ^f:"eg. the median of 2,3,5,7 and 11 is 5, as there are 2 numbers lower, and 2 numbers higher." `q:"If there are an even number of values, there is no middle one. Instead, take the mean of the middle two." c::"eg. the median of 1,4,9 and 16 is (4+9)/2 = 6.5" fscrhdr hstart=300,:endsec        probability  n$="probability" ;"The probability of an event occuring is given by ": "p = N/T where : ": o"N = the number of ways an event can occur. T = the total number of possible events." S:"The probability of picking an ace at random from a full pack of cards is :" ;:"number of aces 4 numbers of cards 52" :"probability = 4/52 = 1/13"  scrhdr  ="The probabilty of an event not occuring is given by : ": w"probability of event occuring = p probability of event not occuring = q ":  "q = 1 - p" L:"In the previous example, the chance of not choosing an ace is :": "p = 1 - 1/13 = 12/13" scrhdr P"Probabilities can be expressed as ratios, decimals or percentages." B:"Certainty is p = 1 or 100% impossibility is p = 0 or 0%" !A:"The chance of a tossed coin coming down heads is : ": $"p(heads) = 1/2 = 0.5 = 50%" &scrhdr (""Total & compound probability": +"Total probability is when one event OR another take place. Thetwo events are mutually exclusive. An example of this isa tossed coin, which can either come down heads or tails." .:"The TOTAL probability that the coin will come down heads or tails is the sum of the two individual probabilities": 0"p = 1/2 + 1/2 = 1" 2scrhdr 5"Compound probability ioccurs when one event AND another take place. An example of this is the probability of two dice thrown together coming down both6s.": 8R"The COMPOUND probability is the product of the two individual probabilities." ::"p = 1/6 x 1/6 = 1/36" J35#,502:0,33!:568,0:0,-33! @S35#+568,502:0,65A:568,0:0,-65A C\35#+568+568,502:0,47/:568,0:0,-47/ F:3,1;"frequency":18,11 ;"marks" G Hscrhdr Jz"The histogram represents the following distribution of marks in maths test given to a class of 18 pupils :" M%:"mark 1 - 10 11 - 20 21 - 30" P "freq. 4 8 6" R:"The bases of the rectangles go from 0.5 to 10.5 and 10.5 to 20.5 etc. so that the enclose the class 1 to 10, 11 to 20 etc.without gaps. " T":"Look at the histogram again." Wscrhdr Z 8204 \scrhdr ^qsthdr a^"Does the frequency go on the :"::"A : vertical scale B : horizontal scale ?" d$s$:s$:s$="A"a$="a"9720% es$"A"s$"a""No, it is A" fZ"Does the class go on the :"::"A : vertical scale B : horizontal scale ?" h$s$:s$:s$="B"s$="b"9720% js$"B"s$"b""No, it is B" k+"Is the frequency represented by the :": n>"A : height of the rectangle B : area of the rectangle." p$s$:s$:s$="B"s$="b"9720% qs$"B"s$"b""No, it is B" rscrhdr tn$="histograms" ustart=800 :endsec    frequency polygons  n$="frequency polygons" "A FREQUENCY POLYGON is a method of showing a frequency distribution. The frequencies are shown on the vertical axis, and the classes on the horizontal axis." @:"An example frequency polygon is shown on the next page."  h=1 scrhdr:920  940 L25,502:180,0:25,502:0,100d .16,3;"0.5 10.5 20.5 30.5" [15,2;"0":13 ,2;"2":11 ,2;"4":9 ,2;"6" -7,2;"8":5,1;"10" Uh=135#,502:0,33!:568,0:0,-33! :63?,83S:119w-63?,115s-83S ^h=135#+568,502:0,65A:568,0:0,-65A  119,115 )119w,115s:568,-18 gh=135#+568+568,502:0,47/:568,0:0,-47/ :3,1;"frequency":18,11 ;"marks"  scrhdr "The frequency polygon represents the following distribution of marks in maths test given to a class of 18 pupils :" %:"mark 1 - 10 11 - 20 21 - 30"  "freq. 4 8 6" :"The bases of the rectangles go from 0.5 to 10.5 and 10.5 to 20.5 etc. so that the enclose the class 1 to 10, 11 to 20 etc.without gaps. " scrhdr::"Notice that the polygon is drawnby joining the mid points of therectangles in a histogram. Normally, frequency polygons aredrawn without the histogram. This is shown on the next page." ,:"Look at the frequency polygon again." scrhdr  h=0  920 scrhdr qsthdr ^"Does the frequency go on the :"::"A : vertical scale B : horizontal scale ?" $s$:s$:s$="A"s$="a"9720% s$"A"s$"a""No, it is A" Z"Does the class go on the :"::"A : vertical scale B : horizontal scale ?" $s$:s$:s$="B"s$="b"9720% s$"B"s$"b""No, it is B" scrhdr n$="frequency polygons" start=900:endsec     frequency distribution  n$="frequency distribution" 1"A frequency distribution is a table showing the number of items in a particular category. The categories, known as CLASSESmay be anything from IQs to shoulder heights of horses - any data showing the number of something in a particular category shows a frequency distribution." scrhdr F:"eg. the frequency distribution of IQs in a grammar school :" &:"IQ 101-120 121-140 141-160" ""Freq. 25% 70% 5%" Q:"eg. the frequency distribution of shoulder heights of horses:" %:"height 14 15 16 17 hands" "freq. 30% 45% 20% 5%" scrhdr "There are several ways of showing a frequency distributionon a form of graph. Two common ways are the HISTOGRAM and the FREQUENCY POLYGON." ?:"There is a section on this tape devoted to each of these."  n$="frequency distribution"  start=1000:endsec K #' $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $ $ $ $getans(loop,return,exp) $ $Udp=0:err=0:u$:u$:(u$)>10 u$="-."u$="-"u$="."err=1 $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $9480%:j $Cerr=1dp>1"type the answer as a number":9460$ $&ans=(u$):ansexp"No, it is ";exp $ans=exp9720% $$egc=egc+1:egc<3loop $return $ % %$(u$(jj))=46.dp=dp+1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1100d:z & 5 crhdT$sthd$ndse$etan$ mh5 EfaDGf6:6:0: maths help tape 5  G C E Tutoring  Feb 1984 !w(20,8):2000 r(10 ) t(5):9069m# Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ fig2=8950":fig=9030F#:axes2=9000(#:eg=9040P#:axes=9100#:triangle=9150#:diagram=218:praise=9720% &squrec=8850":squpar=8900" B:"MATHS help mh5"::"ELEMENTARY & TRANSFORMATIONAL GEOMETRY" :" G C E Tutoring Feb 1984" 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4o:"This tape covers 12 subjects which all come under the headingof elementary & transformationalgeometry." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AF:"Thanks. From now on when I want you to press the C key, I'll say" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"symmetry triangles Pythagoras polygons quadrilaterals rotation translation shearing enlargement stretching angles and angle theorems terms in trans. geometry" K2:"Which subject do you want to try ?":s$: Lc$=s$ N'(s$)>3s$=s$(13) Os$="ang"100d Ps$="sym"200 Qs$="tri" 300, Rs$="Pyt"s$="pyt"400 Ss$="pol"500 Ts$="qua"700 Us$="ter"800  Vs$="tra"900 Ws$="rot"1000 Xs$="enl"1100L Ys$="she"1200 Zs$="str"1400x ^s$"ang"s$"sym"s$"tri"s$"Pyt"s$"pyt"s$"pol"s$"qua"s$"ter"s$"tra"s$"rot"s$"enl"s$="she"s$="str"9200# _ 70F c d f iangles l n!:"Angles and angle theorems": p"Acute angles": so"Acute angles are angles less than 90 (ie. smaller than a right angle). Here are a few examples:": v980P,502:-502,0:502,502 x:150,502:-502,0:502,30 yz=18::z zA"Acute angles which add to make 90 are called COMPLEMENTARY." |scrhdr }"Obtuse angles": m"Obtuse angles are angles greaterthan 90 (ie. greater than a right angle). Here are a few examples:" :80P,65A:-502,0:-20,502 ;200,65A:-502,0:-502,502 z=18::z "Although obtuse angles are greater than 90,they are less than 180. An obtuse angle and an acute angle which add to make180 are called SUPPLEMENTARY." scrhdr :"Reflex angles": k"Reflex angles are angles greaterthan 180. They are the externalangle of an acute or obtuse angle.": ;200,502:-502,0:-502,502 153,502:0,-3:-3,-3:-5,0:-3,3:0,3:3,3:2,0 980P,502:-502,0:502,502 33!,502:0,-3:-3,-3:-5,0:-3,3:0,5:3,3:2,0:2,0:2,-2 z=110 ::z $"Reflex angles are less than 360" scrhdr $"Alternate & corresponding angles" p:"In the diagram below, the anglesa and c are called CORRESPONDINGand angles a and b are called ALTERNATE." '30,100d:100d,0 &30,60<:100d,0 &546,491:60<,60< 10 ,10 ;"a" 13 ,10 ;"b" 15,5;"c" scrhdr qsthdr "An acute angle is :": ^"1. less than 90 2. more than 180 3. between 90 and 180 ?" 4exp=1:egc=3:return=191:getans "An obtuse angle is :": ^"1. less than 90 2. more than 180 3. between 90 and 180 ?" 4exp=3:egc=3:return=194:getans "A reflex angle is :": ^"1. less than 90 2. more than 180 3. between 90 and 180 ?" 4exp=2:egc=3:return=197:getans scrhdr 600X    symmetry  :"Symmetry": q"A symmetrical figure is one that can be divided into pieces which are the mirror image of each other.": Y"The line that separates the mirror image pieces is called the line of symmetry." scrhdr 7"Here is an example of a symmetrical figure:" _30,80P:100d,0:0,502:-100d,0:0,-502 %80P,70F:0,70F z=112 ::z 9"The line through the middle is the line of symmetry." scrhdr "The form of symmetry just described is called BILATERAL symmetry. There is another kind called ROTATIONAL symmetry. For a figure to have rotational symmetry, it must be able to be rotated about its centre but still appear the same.": ["Obviously, all shapes look the same if they are rotated by 360so this does not count."  2n$="symmetry":start=200:return=240 endsec  + , . 1 triangles 4 6!n$="triangles"::"Triangles": 8`"A triangle is a 3 sided figure. The internal angles of a triangle add to make 180.": ;v"There are 3 types of triangle:"::"1. scalene 2. isosceles 3. equilateral" >scrhdr @"scalene triangles": B?"A scalene triangle has all the sides and angles different." EO30,60<:100d,10 :0,60<:-100d,-70F Hz=111 ::z Jg"This is called an obtuse scalenetriangle because one of its internal angles is greater than 90" Lscrhdr O"Isosceles triangle": RD"An isosceles trianglehas two angles and two sides the same.": TQ30,60<:200,0:-100d,60<:-100d,-60< Vz=111 ::z Yd"Notice that there is one line of symmetry - in this case the line of symmetry is vertical" \scrhdr ^"Equilateral triangles": `M"An equilateral triangle has three sides and three angles the same." cO30,60<:100d,0:-502,60<:-502,-60< fz=111 ::z h3"Notice that all the angles are 180/60 ie. 60" jscrhdr mqsthdr p=:"How many angles are the same in a scalene triangle ?" q,exp=0:return=372t:egc=3 rgetans t w@"How many angles are the same in an equilateral triangle ?" z,exp=3:return=382~:egc=3 |getans ~ >"How many angles are the same in an isosceles triangle ?" ,exp=2:return=392:egc=3 getans #start=300,:return=398 endsec      Pythagoras  :"Pythagoras' theorem": ["The square of the hypoteneuse isequal to the sum of the squares of the other two sides." label=0:triangle 5"This applies only to RIGHT ANGLED triangles." scrhdr 2"The theorem can be represented algebraically:" :"AC=AB+BC"  triangle 9"eg. If AB=4 and BC=3 then AC= (4+3)=5" scrhdr c"Pythagoras' theorem allows the calculation of any unknown sidesproviding that two are known.": U"since AC=AB+BC AB=AC-BC BC=AC-AB" :"Some triangles work out so that all the sides are whole numbers.You've already seen one of these(the 3:4:5 triangle).Here are some others:": K"5:12:13 8:15:17 7:24:25" 15,0;"5:12:13 It is worth learning 8:15:17 these as they save time7:24:25 in unnecessary calculation." scrhdr &"Here is an example of a question": !"AB=40cm BC=9cm what is AC ?":  triangle O"AC=AB+BC AC=40+9=1600+81=1681 AC=1681=41" scrhdr qsthdr F"For the triangle below, Pythagoras' theorem states that:": t"1. AB+AC=BC 2. AC=AB+BC 3. AC=AB+BC 4. AC=AB+BC ?"  triangle ,exp=4:egc=3:return=470 getans 4n$="Pythagoras":start=400:return=475 endsec      polygons  :"Polygons": "A polygon is a figure with any number of sides. Four common ones are the triangle (3 sides) quadrilateral (4 sides) hexagon (6 sides) and octagon (8 sides)": p"The most important polygons are the regular ones. A REGULAR polygon has sides all of the same length." scrhdr "Regular polygons":  #"name sides angle":  "Pentagon 5 108 Hexagon 6 120 Heptagon 7 129 Octagon 8 135 Dekagon 10 144" e:"There is a formula for calculating the internal angle of an n-sided polygon. It is:": "angle=90 x (2n-4)/n"  :qsthdr .:"How many sides does a pentagon have ?": ,return=542:egc=3:exp=5 getans ,"How many sides does a octagon have ?": !,return=550&:egc=3:exp=8 $getans &,"How many sides does a heptagon have ?": (,return=558.:egc=3:exp=7 +getans .2n$="polygons":start=500:return=5622 0endsec 2 W X Z ]angle theorems ` b"Angle theorems": d"Theorem 1": gc80P,80P+3,40(:80P-28,80P+3-28:568,0 j'-568,568:0,-568 l(68D,32 :-13 ,-32 n'3,15;"angle x = angle y" q/10 ,7;"x":12 ,13 ;"y" rz=16::z tscrhdr u"Angle theorems" v:"Theorem 2": {80P-3,80P+3,40(:80P-28-3,83S-28:-10 ,42*:64@,-46. ~)-26,34":-28,-28 E10 ,9 ;"o":11 ,6;"x":12 ,9 ;"y" *3,11 ;"angle y = 2(angle x)" F10 ,16;"angle y is at":11 ,16;"the centre." z=16::z scrhdr !"Angle theorems"::"Theorem 3" T80P,78N+15,40(:80P,38&+15:37%,30 )-557,45-:-18,-502 =-(37%-557-18),-(30+45--502) >9 -2,8;"a":15-2,10 ;"c" W12 +1-2,12 +1;"b":12 +1-2,6;"d" \9 -2,16;"Opposite angles":10 -2,16;"add to make 180" V12 -2,16;"a+c=180":12 +1-2,16;"b+d=180" z=14::z q"A quadrilateral drawn so that the corners touch the inside of a circle is called a CYCLIC QUADRILATERAL" scrhdr 8n$="angle theorems":return=678:start=100d endsec     quadrilaterals  :"Quadrilaterals": [:"A quadrilateral is a 4 sided figure. There are 6 main types of quadrilaterals:":: ˲"1. rectangle 2. rhombus 3. kite 4. square 5. parallelogram 6. trapezium": scrhdr "rectangle": P"A rectangle has opposite sides of equal length, and one axis of symmetry." _30,60<:100d,0:0,60<:-100d,0:0,-60< z=110 ::z ;"In this case the axis of symmetry is horizontal." scrhdr  "rhombus": "A rhombus has 4 sides of equal length, but no right angles. It has two lines of symmetry, these being on the diagonals." `30,502:80P,0:20,60<:-80P,0:-20,-60< z=110 ::z 2"A rhombus is a form of parallelogram." scrhdr  "kite": ="A kite has one line of symmetry.Adjacent sides are equal." c30,80P:40(,40(:80P,-40(:-80P,-40(:-40(,40( !z=112 +1::z ;"In this case the axis of symmetry is horizontal." scrhdr  "square": C"A square has 4 sides of equal length, and 4 axes of symmetry." ]30,60<:0,60<:60<,0:0,-60<:-60<,0 z=112 ::z R"The lines of symmetry are horizontal, vertical and the two diagonals." scrhdr  "Parallelogram":  V"A parallelogram has opposite sides of equal length but no lines of symmetry." b30,60<:120x,0:20,60<:-120x,0:-20,-60< z=112 ::z "Opposite angles are equal" scrhdr "Trapezium": 2"A trapezium has one pair of parallel sides" a30,60<:120x,0:20,60<:-80P,0:-60<,-60< z=112 ::z 8n$="quadrilaterals":start=700:return=799 endsec    " %terms in tform geom ( *":"Transformational geometry:": ,"Transformational geometry deals with different ways a geometric figure can be varied mathematically. The operation of changing the geometric figureis called MAPPING, and is done by a MAPPING FUNCTION.": /F"There are six different mapping functions. They are listed below": 2"reflection rotation translation enlargement shear stretch" 4scrhdr 6L"The result produced by the mapping function is called an IMAGE.": 9Z"An image which is the same size and shape as the original figureis called an ISOMETRY." q"These terms are used in the nextsix sections. If you're not surethat you remember them, repeat this section." @ C/n$="terms":start=800 :return=892| zendsec |     translation  :"Translation": n$="translation" l"A translation is the movement ofa figure in any direction. Its size is unaltered, and it is notrotated." eg axes2 x=-6:y=1:fig x=41):y=9 :fig z=17::z scrhdr "A translation is achieved by adding a value to all the x coordinates of a figure, and by adding a value to all the y coordinates." q:"In the previous example the amount added to the x coordinatewas 4, and 1 was added to the y coordinate." /:"This translation is written"::" 4":" 1" /6,47/:0,16,-/4 /17,47/:0,16,/4  :"Look at the diagram again:" scrhdr Z:::"Notice that 4 has been added to the x coordinates, while 1 has been added to y." ?x=41):y=9 :fig:x=-6:y=1:fig:axes2 z=17::z scrhdr #start=900:return=960 endsec      rotation  :"Rotation": n$="rotation" f"In the diagram below the small triangle A has been rotated through 90 to produce triangle C." fig2 z=17::z scrhdr o"The centre of rotation (wheretheimaginary axle is that the diagram has rotated about) is atthe origin." b:" 90 maps (x,y) on to (-y,x) 180 maps (x,y) on to (-x,-y) 270 maps (x,y) on to (y,-x)" fig2 z=15::z \"A rotation of 90 means 90 anticlockwise. Negative rotations are clockwise." scrhdr  %start=1000:return=1085= :endsec = K L N Q enlarge T V:"Enlarge": Wn$="enlarge" XZ"In the diagram below the small triangle A has been enlarged to produce triangle B." ^fig2 _z=17::z `scrhdr bq"To enlarge a figure by a certainamount (the SCALE FACTOR), multiply the coordinates by the scale factor." cX"In the diagram below all the coordinates of triangle A have been multiplied by 2" efig2 hz=17::z jscrhdr l%start=1100L:return=1148| yendsec |    shear   :"Shear":  n$="shear" b"In the diagram below the rectangle ABCD has been sheared to produce parallelogram EBCF." squpar  H"Notice that all the lines of therectangle have changed except one." scrhdr %start=1200:return=1230 endsec   x z } stretch  :"Stretch": n$="stretch" `:"A stretch is an enlargement in one axis or direction only. The axis can be either x or y." squrec  H"The square ABCD has been stretched to form the rectangle ABEF" scrhdr G:"The stretch in this section was by a factor of 2 in the x axis.": %start=1400x:return=1435 endsec      reflection  :"Reflection" n$="reflection" :"Reflection": "Reflection in geometry is the production of an mirror image of the original figure. The imaginary mirror is at the LINE OF REFLECTION." F:"The axis of reflection can be any line, eg. x axis, y axis etc" scrhdr %start=1500:return=1528 endsec  x$="": o=1̱x$ n=18 #w(o,n):x$(oo)+n-1,w(o,n) n o  H24,16,16,32 ,16,16,24,0 C24,8,8,4,8,8,24,0 F0,0,62>,64@,64@,64@,62>,0 E0,0,124|,2,2,2,124|,0 G0,66B,66B,66B,66B,66B,44,,0 G0,44,,66B,66B,66B,66B,66B,0  G0,0,60<,64@,124|,64@,60<,0 I240,8,120x,128,248,0,0,0 C0,0,0,4,126~,4,0,0 I248,136,136,136,0,0,0,0 C4,4,228,4,0,0,0,0 G240,8,240,8,240,0,0,0 G28,20,16,16,8,8,40(,568  E4,68D,228,68D,0,0,0,0 "G16,16,16,0,0,480,72H,480 %E12 ,18,18,12 ,0,0,0,0 (E7,4,4,8,8,144,80P,32 4 )::(40(*3.14159Iρ/180)  z=31000 %v=((z^2-(z-1)^2)) u=v^0.5 C((u)-(u+0.5))<0.01z# =z;":";z-1;":";v^0.5 z  " "square + rectangle "]27,568:502,0:0,502:-502,0:0,-502 "_27,568:100d,0:0,502:-100d,0:0,-502 "$7,2;"A D F" "%15,2;"B C E" " " "square + parallelogram "]27,568:75K,0:0,502:-75K,0:0,-502 "`27,568:75K,0:25,502:-75K,0:-25,-502 "#7,3;"A E D F" "!15,3;"B C" " " #K27,568:110n,0:568,41):0,557 #f15,7;"0":15,9 ;"2":15,11 ;"4":15,12 +1;"6" #E14,6;"0":12 ,6;"2":10 ,6;"4" #K27,568:110n,0:568,41):0,557 # [568+16,568+8:0,16:8,-16:-8,0 # ^568+32 ,568+16:0,32 :16,-32 :-16,0 #\568-10 ,568+16:-16,0:16,8:0,-8 #11 ,9 ;"A" #9 ,12 ;"B" #11 ,4;"C" #& #' #( #2K27,41):110n,0:27,41):0,557 #:]17,3;"0":17,6;"2":17,9 ;"4":17,12 ;"6" #<17,15;"8" #>\16,2;"0":14,2;"2":12 ,2;"4":10 ,2;"6" #A #Fi579+x,513+y:20,20:-20,20:-20,-20:20,-20: #Hc72H+x,568+y:0,30:-30,0:0,-30:30,0: #P'"Here is an example of ";n$ #R #Z #\ #_noises #b #d-z=120:0.01z# =,-24:z #f #h #i&z=15:0.3,t(z):z #l #mz=15:t(z):z: #n*7,9 ,5,-7,0 # # # #axes # #_30,90Z:210,0:30,90Z:0,557:0,-110n #10 ,2;"0" #f11 ,9 ;"90":11 ,15;"180":11 ,22;"270":11 ,29;"360" #atype=09124# #atype=19130# #.4,2;"1":16,1;"-1" #07,1;".5":13 ,0;"-.5" # #.7,2;"1":13 ,1;"-1" #.4,2;"2":16,1;"-2" # # # #draw triangle # #N30,60<:100d,0:0,60<:-100d,-60< #label=09175# #15,6;"adjacent" # 8,1;"hypoteneuse" #11 ,18;"opposite" #6123{,60<:0,7:7,0 #H41),60<:0,2:-3,2:-2,0 #14,2;"A" #14,17;"B" #6,17;"C" #::::::::: # # #':9200#+((*6)+1): #"Berk ! What's ";c$;" ?": #'"Wally ! ";c$;" isn't on the tape.": ##"Moron ! Try spelling straight": #"sorry, ";n$;" ? ": #"Are you tired, ";n$;" ?": #"Don't wind me up, ";n$: $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $?dp=0:err=0:u$:u$:(u$)>10 err=1 $(u$)=09460$ $]j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $%9480%:j:u$="-."err=1 $Cerr=1dp>1"type the answer as a number":9460$ $=ans=(u$):(exp-ans)>0.005y# ="No, the answer is ";exp $$(ans-exp)<0.005y# =9720% $return $ % %$(u$(jj))=46.dp=dp+1 % ,j>1Ư(u$(jj))=45-err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z & % >@@@>||BBBBB,,BBBBB<@|@<x~(8DD0H0  P X 5  m#crhdT$sthd$ndse$etan$ig"iF#xes(#P#xe#riangl#iagrarais%qure"qupa" mh6 zVuz6:6:0: maths help tape 6  G C E Tutoring  Feb 1984 !w(25,8):2000 r(10 ) t(5):9069m# Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ fig2=8950":fig=9030F#:axes2=9000(#:eg=9040P#:axes=9100#:triangle=9150#:diagram=218:praise=9720% 5br=8100:squrec=8850":squpar=8900" ,:"MATHS help mh6"::"MATRICES & VECTORS" :" G C E Tutoring Feb 1984" 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4^:"This tape covers 9 subjects which all come under the headingof matrices and vectors." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AF:"Thanks. From now on when I want you to press the C key, I'll say" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"arithmetic of matrices determinants inverse matrices solving equations with matrices matrix transposes transformational matrices scalars and vectors vector arithmetic & laws velocity triangles" K2:"Which subject do you want to try ?":s$: Lc$=s$ N'(s$)>3s$=s$(13) Os$="ari"600X Ps$="det"200 Qs$="inv" 300, Rs$="sol"400 Ss$="mat"500 Ts$="tra"900 Us$="pro"800  Vs$="sca"1700 Ws$="vec"1800 Xs$="law"1100L Ys$="vel"1900l ^^s$"mat"s$"det"s$"inv"s$"sol"s$"tra"s$"ari"s$"sca"s$"vec"s$"vel"9200# _ 70F c d f    determinants  :"Determinants" X:"The determinant of a matrix is a value which can be calculated from its elements." D:"Here is how to calculate the determinant of a 2 x 2 matrix:" *:" a b c d" q42*-38&,80P:0,16,-/3:75K-38&,80P:0,16,/3 :"determinant = ad-bc" scrhdr *:" a b c d" 42*-38&,104h+40(:0,16,-/3:75K-38&,104h+40(:0,16,/3 :"determinant = ad-bc" :"so the determinant of: 1 2 3 4 is (1 x 4)-(2 x 3) = -2" q42*-38&,87W:0,16,-/3:75K-38&,87W:0,16,/3 :qsthdr *:" a b c d" 42*-38&,104h+47/:0,16,-/3:75K-38&,104h+47/:0,16,/3 :"determinant = ad-bc" @:" ";r(1);" ";r(2):" ";r(3);" ";r(4) 42*-38&,104h+7:0,16,-/3:75K-38&,104h+7:0,16,/3 .:"what is the determinant of this matrix ?" @:exp=r(1)*r(4)-r(2)*r(3):egc=3 return=250 getans !:"more questions ?"::a$:a$ $a$(11)="y"232 scrhdr %n$="determinants":return=260 endsec  s42*,104h-579:0,16,-/3:75K,104h-579:0,16,/3 + , . 1 inverses 6:"Inverses": 8"The inverse of a matrix A is written A. When the inverse is multiplied by the original matrix, the identity or unit matrix is produced.": ;l"This can be thought of as being similar to multiplying a number by its reciprocal and obtaining unity.": >"so A x A = I": @scrhdr B["Finding the inverse of a matrix is relatively simple. It consists of 3 stages:": E"1. swap the top left and lower right elements 2. change the signs of the lower left and top right elements 3. divide by the determinant": Fw"So to find the inverse of "::" 7 2 1 4"::"for example, use the following 3 stages:" G'x=8:y=546:i=24:br Hscrhdr J7"1. swap the top left and lower right elements": Li"so"::" 7 2 1 4"::"becomes"::" 4 2 1 7": M'x=8:y=71G:i=24:br Ny=111o:br Oscrhdr RF"2. change the signs of the top right and lower left elements": T="so"::" 4 2":" 1 7"::"becomes"::" 4 -2":" -1 7": U br:y=70F:i=40(:br Vscrhdr Y""3. divide by the determinant": Z?"since the determinant is"::"(4 x 7)-(2 x 1) = 28-2 = 26": \["so"::" 4 -2":" -1 7"::"becomes"::"1/26 4 -2 -1 7": ],y=86V:br:y=480:x=39':br ^4"which is the inverse of the original matrix." `scrhdr 2start=300,:return=398:n$="inverses" endsec     solving equations  6:"Solving simultaneous equations using matrices": f"This is an alternative to the conventional method of substitution. It has six stages:": "1. Write down the equations 2. put them in matrix form 3. find the matrix's inverse 4. multiply by the inverse 5. simplify the equation 6. evaluate the final matrix" ,:"Let's take these stages one by one:": scrhdr  "1. write down the equation": ="Here's an example of a simultaneous equation :": 2"7x + 2y = 33 x + 4y = 27": scrhdr "2. put them in matrix form" O:"The coefficients (numbers beforethe variables) go into a 2x2 matrix:": "x and y make another matrix, and the values of the two equations a third matrix. This is how it looks written out:": 0" 7 2 x = 33 1 4 y 27" _x=8:y=63?:i=24:br:x=45-:i=12 :br:x=79O:i=18:br &:"Compare this with the original": 2"7x + 2y = 33 x + 4y = 27": scrhdr /"3. find the inverse of the 2x2 matrix": "The inverse of 7 2 1 4 is 1/26 4 -2 -1 7" (x=8:y=111o:i=24:br (x=568:y=71G:i=41):br F:"If you didn't follow that then look at the section on inverses." scrhdr "4. multiply by the inverse" 2:" 7 2 x = 33 1 4 y 27" `x=8:y=136:i=24:br:x=45-:i=12 :br:x=78N:i=18:br :"becomes:" ::"1/26 4 -2 7 2 x -1 7 1 4 y": ax=43+:y=96`:i=36$:br:x=95_:i=24:br:x=134:i=12 :br 6:"= 1/26 4 -2 33 -1 7 27" Ex=58::y=64@:i=37%:br:x=110n:i=18:br scrhdr "5. simplifiy the equation": f"Any matrix multiplied by its inverse produces the identity matrix, so the equation becomes:": ?:" 1 0 x = 1/26 4 -2 33 0 1 y -1 7 27" W:"and the identity matrix doesn't affect another matrix, so the equation becomes:" }x=8:y=95_:i=24:br:x=44,:i=12 :br:x=128:i=40(:br:x=182:i=17:br ::" x = 1/26 4 -2 33 y -1 7 27" `x=6:y=38&:i=12 :br:x=89Y:i=38&:br:x=144:i=16:br scrhdr "6. evaluate the result": ::" x = 1/26 4 -2 33 y -1 7 27" `x=7:y=126~:i=8:br:x=89Y:i=38&:br:x=143:i=16:br E:" x = 1/26 132-54 1/26 78 = 3 y -33+189 156 6": "so x=3 and y=6" |x=7:y=102f:i=8:br:x=73I:i=579:br:x=179:i=25:br:x=223:i=9 :br Jn$="solving equations using matrices":return=499:start=400  :endsec     transpose  :"Transpose": O"The transposition of a matrix isa very simple operation. Here isan example:" /:" B = 1 2 3 4" a42*,104h:0,16,-/3:75K,104h:0,16,/3 K:"To transpose B, change the rows for columns and the columns for rows." /:" B'= 1 3 2 4"  s42*,104h-579:0,16,-/3:75K,104h-579:0,16,/3  scrhdr  qsthdr : O"What do you interchange with the rows of a matrix to make itstranspose ?"   a$:a$ 8a$="columns"a$="the columns"a$="The columns"praise da$"columns"a$"the columns"a$"The columns""No, you interchange the rows with the columns" 3start=500:return=550&:n$="transpose" endsec & W X Z ]arithmetic of matrices ` b"Matrix arithmetic": dZ"A matrix is a way of arranging an array of numbers. Here is an example of a matrix:": g" 2 1":" 4 3": i(x=8:y=112p:i=24:br j:"It is easy to add matrices together. All you have to do is add the ELEMENTS (the individualnumbers) to the corresponding ones.": lD" 2 1 + 3 2 = 2+3 1+2 = 5 3":" 4 3 1 0 4+1 3+0 5 3" mqx=8:y=32 :i=24:br:x=71G:br:x=136:i=557:br:x=214:i=26:br nscrhdr q"Multiplying matrices is harder. The easiest case is if one of the matrices is a COLUMN matrix (a matrix only 1 number wide)."::"Here's an example:": tE" 2 1 x 3 = 3x2 + 5x1 = 11 4 3 5 4x3 + 3x5 27": u|x=8:y=95_:i=24:br:x=71G:i=8:br:x=118v:i=75K:br:x=231:i=15:br v}"This is a bit hard to follow at first. The general example of multiplying a matrix by a columnmatrix looks like this:": {:" a b x e = ea+fb c d f ec+fd": |_x=8:y=32 :i=24:br:x=71G:i=8:br:x=121y:i=37%:br ~scrhdr u"Multiplying two square matrices together (matrices with the samenumber of columns and rows) is trickier still:": +"The general example looks like this:": >" a b x e g = ea+fb ga+hb c d f h ec+fd gc+hd": `x=8:y=86V:i=24:br:x=546:i=25:br:x=102f:i=90Z:br "The first column of the result is the same as in the last example. The second column is similar except g and h have beenused instead of e and f." scrhdr  8200 "It is only possible to multiply two matrices together if the number of columns of the LEFT matrix is the same as the numberof rows of the RIGHT matrix."  qsthdr :" a b x e = ea+fb c d f ec+fd": :: g" ";r(1);" ";r(2);" x ";r(5);" = ?":" ";r(3);" ";r(4);" ";r(6) `x=8:y=158:i=24:br:x=71G:i=8:br:x=121y:i=37%:br Dx=8:y=112p:i=24:br:x=546:i=10 :br a:"Type the top number, then press enter, then type the bottom number, then enter again.": Xreturn=668:exp=r(1)*r(5)+r(2)*r(6):egc=3:getans Xreturn=670:exp=r(3)*r(5)+r(4)*r(6):egc=3:getans "More questions ?":a$:a$ $a$(11)="y"655      " ( z;start=600X:return=898:n$="matrix arithmetic" endsec     trans. matrices   translation  :"translation" x:"In transformational geometry, a translation is a movement of a geometric figure without changing its shape" axes2 x=-6:y=1:fig x=41):y=9 :fig z=17::z scrhdr "A translation is achieved by adding a value to all the x coordinates of a figure, and by adding a value to all the y coordinates." q:"In the previous example the amount added to the x coordinatewas 4, and 1 was added to the y coordinate." scrhdr /:"This translation is written"::" 4":" 1" 86,47/+80P:0,16,-/4 817,47/+80P:0,16,/4 y:"This matrix will translate any coordinate of an original geometric figure if it is added to that coordinate."  :"Look at the diagram again:" scrhdr Z:::"Notice that 4 has been added to the x coordinates, while 1 has been added to y." ?x=41):y=9 :fig:x=-6:y=1:fig:axes2 z=17::z scrhdr scrhdr    rotation  :"Rotation": 8000@:scrhdr f"In the diagram below the small triangle A has been rotated through 90 to produce triangle C." fig2 z=17::z scrhdr  8010J o"The centre of rotation (wheretheimaginary axle is that the diagram has rotated about) is atthe origin." b:" 90 maps (x,y) on to (-y,x) 180 maps (x,y) on to (-x,-y) 270 maps (x,y) on to (y,-x)" fig2 z=15::z \"A rotation of 90 means 90 anticlockwise. Negative rotations are clockwise." scrhdr  *"The other matrices for rotation are:": '"for 180:"::"-1 0":" 0 -1": <"for 270:"::" 0 1 -1 0": $scrhdr L N Q enlarge T V:"Enlarge": XZ"In the diagram below the small triangle A has been enlarged to produce triangle B." ^fig2 _z=17::z `scrhdr bq"To enlarge a figure by a certainamount (the SCALE FACTOR), multiply the coordinates by the scale factor." cX"In the diagram below all the coordinates of triangle A have been multiplied by 2" efig2 hz=17::z jscrhdr l8"The transformation matrix to usefor enlargement is": o" k 0":" 0 k": rD"where k is the scale factor. In this case the matrix would be:": t" 2 0":" 0 2": vp"The original coordinates are multiplied by the transformationmatrix to produce the new coordinates." ~scrhdr   shear   :"Shear": b"In the diagram below the rectangle ABCD has been sheared to produce parallelogram EBCF." squpar  H"Notice that all the lines of therectangle have changed except one." scrhdr p"The transformation matrix for shearing varies according to thedegree of shear. The general matrix is:": " 1 a":" 0 1": X"Multiplying the coordinates by the transformation matrix gives the new coordinates." F:"The larger the value of a, the greater the shear in the x axis." scrhdr x z } stretch   "Stretch": `:"A stretch is an enlargement in one axis or direction only. The axis can be either x or y." squrec  H"The square ABCD has been stretched to form the rectangle ABEF" scrhdr G:"The stretch in this section was by a factor of 2 in the x axis.": scrhdr ^"The transformation matrix for a stretch is similar to the matrix for shear. It is:": " a 0":" 0 1": "Multiplying by the transformation matrix produces the new coordinates. The larger the value of a the larger the stretch in the x axis" scrhdr  1600@    @ B E identity H J"Identity": L"The identity matrix produces no change when multiplied by the coordinates of a geometric figure. This is not suprising as any matrix is unchanged when multiplied by the identity matrix." r3n$="identity":start=900:return=1660| wendsec |    vectors  :"Scalars and vectors": "Quantities such as mass, volume and length are called SCALAR quantities. Scalar quantites have size but not direction.": o"Quantities such as velocity have both size and direction. This type of quantity is called a VECTOR.": D"The general property of a vectoris that it has two components.": scrhdr "Position vectors": "Since a vector has two components, it can represent a position by taking the value of x and y coordinates. This is written as follows:": " 2":" 1": 'x=7:y=87W:i=10 :br -"The x coordinate is the top number.": scrhdr "A position vector represents an increment to existing coordinates NOT the actual coordinates of a point.": :"In the above example the vectormeant ""move 2 in the x directionand 1 in the y direction.""" qsthdr 1:"Which of these is a vector quantity ? " S:"1. length 2. time 3. veolcity": 5exp=3:egc=3:return=1760:getans P:"Is the x coordinate of a position vector on the top or bottom ?":  a$:a$ a$="top"praise %a$"top""No, it's the top figure" 3start=1700:return=1798:n$="vectors" endsec       vector arith. & laws  ":"Vector arithmetic and laws": ^"Vector arithmetic is very simpleand follows the laws of conventional arithmetic.": "Addition": <"Two vectors a and b add to make a vector c as follows:": ""a = 2 b = 3":" 1 4" 8i=12 :x=29:y=63?:br:x=93]:br &:"c = 2+3 = 5":" 1+4 = 5" x=101e:y=39':br  x=39':i=24:br !scrhdr $""a = 2 b = 3":" 1 4" &&:"c = 2+3 = 5":" 1+4 = 5" '9i=12 :x=29:y=151:br:x=93]:br (n:"Vector addition is commutative since it makes no difference which way round the vectors are added.": )&x=101e:y=39'+88X:br *x=39':i=24:br +"a + b = b + a = c": ."This is easy to see if you thinkof the vectors as position vectors. It makes no difference which one is taken first - the final position is the same." 0scrhdr 2"vector addition is also associative. Three vectors a, b and c add to make the same result in both of the following ways:": 54" d = a + (b + c) d = (a + b) + c" 8_:"Again, this can be easily understood if the vectors are thought of as positional." :3:"Vector subtraction is also conventional." <scrhdr ?"Vector multiplication": B/"A vector can be multiplied by a scalar:": D1" eg. a = 2 1": F:" eg. 2a = 2 x 2 = 4 2 x 1 = 2" GFi=12 :x=165:y=87W:br:x=77M:y=111o:br H(i=44,:x=85U:y=87W:br Iscrhdr Lqsthdr N/:"Is vector addition commutative ?":a$:a$ P;a$="No"a$="no""Yes it is ! Read the section again." Sa$="yes"a$="Yes"praise V(:"Is vector addition associative ?": X a$:a$ Z;a$="No"a$="no""Yes it is ! Read the section again." ]a$="yes"a$="Yes"praise b=n$="vector arithmetic":start=1800:return=1898j gendsec j k l n qvelocity triangles t v:"Velocity triangles"::"Sometimes it is necessary to know the length of a vector rather than its individual components.": x"Since the two components of a positional vector are at right angles to each other, the lengthof a vector can be calculated using Pythagoras' theorem." yscrhdr {:"For example, the length of a vector:"::" 4":" 3":::::::::"is 5. This can be seen in the triangle above." }1x=14-1:y=119w:i=12 :br ~label=0:triangle O"AB represent a the x component and BC the y component. AC is the length." scrhdr "The length of the vector can represent quantites such as speed. If an aircraft is flying at right angles to the wind its ground speed can be calculated using vectors:":  triangle q"AB represents the aircraft's airspeed, BC represents the wind velocity and AC represents the ground speed." 8,17;"air speed=150":10 ,17;"wind speed=80":12 ,17;"AC=150+80":12 +1,20;"AC=28900":14,21;"AC=170"  :::: scrhdr "Sometimes velocities will not be at right angles. In these cases Puthagorus cannot be used as the resultant triangle is not right angled." d:"The length can be calculated in these cases by drawing the triangle and measuring length." >n$="velocity triangles":start=1900l:return=1998 endsec   x$="": o=1̱x$ n=18 #w(o,n):x$(oo)+n-1,w(o,n) n o  H24,16,16,32 ,16,16,24,0 C24,8,8,4,8,8,24,0 F0,0,62>,64@,64@,64@,62>,0 E0,0,124|,2,2,2,124|,0 G0,66B,66B,66B,66B,66B,44,,0 G0,44,,66B,66B,66B,66B,66B,0  G0,0,60<,64@,124|,64@,60<,0 I240,8,120x,128,248,0,0,0 C0,0,0,4,126~,4,0,0 I248,136,136,136,0,0,0,0 C4,4,228,4,0,0,0,0 G240,8,240,8,240,0,0,0 G28,20,16,16,8,8,40(,568  E4,68D,228,68D,0,0,0,0 "G16,16,16,0,0,480,72H,480 %E12 ,18,18,12 ,0,0,0,0 (E7,4,4,8,8,144,80P,32 *G4,8,16,16,16,32 ,32 ,32 ,G32 ,32 ,32 ,16,16,16,8,4 /C32 ,16,8,8,8,4,4,4 2C4,4,4,8,8,8,16,32 4 )::(40(*3.14159Iρ/180)  z=31000 %v=((z^2-(z-1)^2)) u=v^0.5 C((u)-(u+0.5))<0.01z# =z;":";z-1;":";v^0.5 z  z=07 z:7-z 7x=13:"" c=110 :c x z 0::" G C E TUTORING"::" Orders"  @"A geometric figure can be rotated by multiplying the individual coordinates by one ofthree different transformation matrices, depending on whether rotation through 90,180 or 270 is required.": B&"For a 90 rotation, multiply by:": E" 0 -1":" 1 0": H,"A 90 rotation is shoen in the diagram." I J:"The coordinates of the corners of the triangle are:": L&" 2 2 3":" 2 1 1": MM"which when multiplied by the matrix for a 90 rotation become:": N?" 0 -1 2 0-3 -3 1 0 3 2+0 2": O?" 0 -1 2 0-1 -1 1 0 1 2+0 2": P=" 0 -1 3 0-1 -1 1 0 1 3-0 3" R scrhdr:  Gx,y:0,16,-/3:x+i,y:0,16,/3: "There is a special matrix calledthe UNIT or identity matrix. If another matrix is multiplied by it there is no effect, rather like multiplying a number by 1 in ordinary arithmetic." $:"This is the 2x2 unit matrix:": " 1 0":" 0 1": 'x=8:i=24:y=80P:br scrhdr  " "square + rectangle "]27,568:502,0:0,502:-502,0:0,-502 "_27,568:100d,0:0,502:-100d,0:0,-502 "$7,2;"A D F" "%15,2;"B C E" " " "square + parallelogram "]27,568:75K,0:0,502:-75K,0:0,-502 "`27,568:75K,0:25,502:-75K,0:-25,-502 "#7,3;"A E D F" "!15,3;"B C" " " #K27,568:110n,0:568,41):0,557 #f15,7;"0":15,9 ;"2":15,11 ;"4":15,12 +1;"6" #E14,6;"0":12 ,6;"2":10 ,6;"4" #K27,568:110n,0:568,41):0,557 # [568+16,568+8:0,16:8,-16:-8,0 # ^568+32 ,568+16:0,32 :16,-32 :-16,0 #\568-10 ,568+16:-16,0:16,8:0,-8 #11 ,9 ;"A" #9 ,12 ;"B" #11 ,4;"C" #& #' #( #2K27,41):110n,0:27,41):0,557 #:]17,3;"0":17,6;"2":17,9 ;"4":17,12 ;"6" #<17,15;"8" #>\16,2;"0":14,2;"2":12 ,2;"4":10 ,2;"6" #A #Fi579+x,513+y:20,20:-20,20:-20,-20:20,-20: #Hc72H+x,568+y:0,30:-30,0:0,-30:30,0: #P'"Here is an example of ";n$ #R #Z #\ #_noises #b #d-z=120:0.01z# =,-24:z #f #h #i&z=15:0.3,t(z):z #l #mz=15:t(z):z: #n*7,9 ,5,-7,0 # # # #axes # #_30,90Z:210,0:30,90Z:0,557:0,-110n #10 ,2;"0" #f11 ,9 ;"90":11 ,15;"180":11 ,22;"270":11 ,29;"360" #atype=09124# #atype=19130# #.4,2;"1":16,1;"-1" #07,1;".5":13 ,0;"-.5" # #.7,2;"1":13 ,1;"-1" #.4,2;"2":16,1;"-2" # # # #draw triangle # #N30,60<:100d,0:0,60<:-100d,-60< #label=09175# #15,6;"adjacent" # 8,1;"hypoteneuse" #11 ,18;"opposite" #6123{,60<:0,7:7,0 #H41),60<:0,2:-3,2:-2,0 #14,2;"A" #14,17;"B" #6,17;"C" #::::::::: # # #':9200#+((*6)+1): #"Berk ! What's ";c$;" ?": #'"Wally ! ";c$;" isn't on the tape.": ##"Moron ! Try spelling straight": #"sorry, ";n$;" ? ": #"Are you tired, ";n$;" ?": #"Don't wind me up, ";n$: $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $?dp=0:err=0:u$:u$:(u$)>10 err=1 $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $%9480%:j:u$="-."err=1 $Cerr=1dp>1"type the answer as a number":9460$ $=ans=(u$):(exp-ans)>0.005y# ="No, the answer is ";exp $$(ans-exp)0.005y# =9720% $return $ % %$(u$(jj))=46.dp=dp+1 % ,j>1Ư(u$(jj))=45-err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z &  >@@@>||BBBBB,,BBBBB<@|@<x~(8DD0H0  P    X 5  m#crhdT$sthd$ndse$etan$ig"iF#xes(#P#xe#riangl#iagrarais%qure"qupa"' mh7 IFnB0KF6:6:0: maths help tape 7  G C E Tutoring  Jan 1984 !w(20,8):2000 r(10 ) Lscrhdr=9300T$:qsthdr=9350$:endsec=9400$:getans=9450$ &diagram=218:praise=9720% ":"MATHS help mh7"::"CALCULUS" :" G C E Tutoring Jan 1984" 2l:"Hello. What's your name ?":n$:n$:"OK, ";n$;", I'll begin by telling you how to use this maths tape" 4R:"This tape covers 8 subjects which all come under the headingof calculus." :>:"Press the key marked C so I can type the next screenfull" <a$=:a$="c"62> = 60< >:z=1100d:z AG:"Thanks. From now on when I want you to press the C key, I'll say:" D:" Hit C to continue" F1:"Here's the list of subjects on this tape": H"differentiation turning points maxima minima rates of change velocity acceleration integration" K2:"Which subject do you want to try ?":s$: Lc$=s$ N'(s$)>3s$=s$(13) Os$="dif"100d Ps$="tur"200 Qs$="max" 300, Rs$="min"400 Ss$="rat"600X Ts$="vel"700 Us$="acc"800  Vs$="int"900 ^Us$"dif"s$"tur"s$"max"s$"min"s$"rat"s$"vel"s$"acc"s$"int"9200# _ 70F c d f idifferentiating l n:"Differentiation": p:"Finding the gradient of a curve at a given point can be done in two ways - drawing a graph and measuring it, or using a method known as differentiation. " s:"Differentiation has two advantages over the graphical method. Firstly, it is quicker for most functions. Secondly, it is totally accurate, unlike graphs, which are only as accurate as you draw them." vscrhdr z"Suppose you need to find the gradient at some point on the line y = x. If you worked it out graphically you would need to know two things:"::"1. f(x) 2. f(x+h)": }j"where x is the point that the gradient is to be calculated, and x+h is a little bit further along." scrhdr "The smaller h is, the nearer youget to measuring the gradient atthe exact point, but as h gets smaller, it gets harder to measure accurately.": "Differentiation allows the gradient to be calculated algebraically. This is shown below for the function f(x)=x:": scrhdr "gradient = (f(x+h)-f(x))/h = ((x+h) - x)/h = (x+2xh+h-x)/h = (2xh+h)/h = 2x+h" p:"If h is made very small, the gradient tends to the value 2x. Expressed algebraically, this becomes:": "If h0 then 2x+h2x" ':"The symbol means ""tends to "" " scrhdr ?"The symbol for the gradient of the tangent is dy/dx, so:": "if y=x then dy/dx=2x" (:"This can be written differently:": "if y=x then dy/dx=nx" :"so if y=x then dy/dx=3x" V:"This process of finding nx from x is called differentiation."  qsthdr  "If y=5x is dy/dx=15x ?" &s$:s$:s$="yes"s$="y"9720% s$"yes"s$"y""Yes it is !" :"If y=6x is dy/dx=6 ?" &s$:s$:s$="yes"s$="y"9720% s$"yes"s$"y""Yes it is !" scrhdr d"If dy/dx is differentiated, the result is called dy/dx. This result is used in later sections" scrhdr 9start=100d:return=198:n$="differentiation" endsec     turning points  :"Turning points": Ԍ"A turning point is the point on a graph where the curve changes direction from upwards to downwards, or from downwards to upwards." -:"This is shown on the diagram below :" 218 225 K25,25:120x,0:25,25:0,60< M40(,40(:30,30:30,-30:30,30 /12 ,9 ;"A":17,12 ;"B"  12 ,21;"The points":13 ,21;"marked A&B":14,21;"are turning":15,21;"points" :::scrhdr |"At the exact point of turning, the gradient of the curve is 0. Since the gradient is given by dy/dx, it follows that :" ::"At the turning point dy/dx = gradient = 0" y:"This makes it very easy to find the turning points of a given function. Simply solve the equation dy/dx = 0." 0:"An example of this is on the next page." :scrhdr /"where is the turning point on the curve :" :" 2":"y = x - 4x" ,:"First find dy/dx :"::"dy/dx = 2x - 4" :"Now solve for dy/dx = 0": "2x - 4 = 0 so x = 2" R:"find y by putting the value for x back into the original equation :": $"y = 4 - 4(2) = -4 so t.p. (2,-4)" scrhdr qsthdr  {:"To find a turning point, what value do you solve dy/dx for ?":exp=0:loop=268 :return=268 :getans  @n$="turning points":return=270:start=200:endsec  + , . 1maxima 4 6:"There are two types of turning points: maxima and minima. To determine which type a turning point is, differentiate the equation for dy/dx. If the result is negative, then the turning point is a maximum." 8#:" 2":"eg. y = -x - 4" ;:"dy/dx = -2x " >:" 2 2":"d y/dx = -2" @9:"So in this case the turning point is a maximum." Bscrhdr C::"If a turning point is a maximum,it represents the maximum value that the function reaches beforediminishing. On the diagram below, point A is a maximum." E diagram Hscrhdr Jqsthdr L:"The result obtained by differentiating dy/dx allows you to distinguish between a maximum and a minimum. Is it negative or positive for a maximum ? " O :s$:s$ R+s$="negative"s$="-ve"s$="-"9720% T6s$"negative"s$"-ve"s$"-""No, it is negative." V0start=300,:return=348\:n$="maxima" Yendsec \   minima  :"There are two types of turning points: maxima and minima. To determine which type a turning point is, differentiate the equation for dy/dx. If the result is positive, then the turning point is a minimum." !:" 2":"eg. y = x - 4" :"dy/dx = 2x " :" 2 2":"d y/dx = 2" 9:"So in this case the turning point is a minimum." scrhdr ::"If a turning point is a minimum,it represents the minimum value that the function reaches beforediminishing. On the diagram below, point B is a minimum."  diagram scrhdr qsthdr :"The result obtained by differentiating dy/dx allows you to distinguish between a maximum and a minimum. Is it negative or positive for a minimum ? "  :s$:s$ +s$="positive"s$="+ve"s$="+"9720% 6s$"positive"s$"+ve"s$"+""No, it is positive." 0start=400:return=448:n$="minima" endsec       W X Z ]rates of change ` b:"Rates of change": d"The gradient of a line represents the rate at which y changes as x is varied. If the x axis represents time, then the gradient can measure physical quantities, such as velocity or acceleration." gb:"Differentiation makes it possible to calculate these quantities algebraically. ": hscrhdr jqsthdr l[:"To allow the calculation of a rate of change, what must the x axis represent ? ": n s$:s$ qs$="time"praise r t9start=600X:return=632x:n$="rates of change" vendsec x     velocity   "Velocity" y:"The rate of change of distance with time is written ds/dt, and is calculated by differentiatinga function s=f(t)": :"eg. s=t+3t+9 ds/dt=2t+3 so to calculate the velocity at t=3 seconds, for example, put 3 into the equation ds/dt=2t+3 ds/dt=2(3)+3=9ms" scrhdr qsthdr U:"if s=";r(1);"t+";r(3);" what is the":"velocity at t=2 seconds ?": $"I'll help you a bit with this:": "what is ds/dt ?"  s$:s$ <s$="velocity""No, I mean what's the equation":730  (s$)7s$=s$(7) 3s$(11)=(r(1)*2)praise (:"OK, so put t=2 into the equation": "What's the value of ds/dt ?" 7exp=r(1)*4:egc=3:return=748 getans scrhdr $"Would you like another question?"  s$:s$ =s$="yes""Yes what ?":s$:s$:s$"yes please"750 $s$(11)="y"720 $s$(11)="n"760 &"Please answer yes or no":748 2start=700:return=765:n$="velocity" endsec     " %acceleration ( *:"Acceleration": ,u"The rate of change of velocity with time is written dv/dt, and is calculated by differentiatinga function v=f(t)" /I:"eg v=9.8t dv/dt=9.8 Notice that in this example, although velocity depends on time, acceleration doesn't. You can tell this because the equation involving dv/dt does not have a term including t in it." 2scrhdr 4qsthdr 6Y:"if v=";r(1);"t+";r(3);" what is the":"acceleration at t=2 seconds ?": 9$"I'll help you a bit with this:": <"what is dv/dt ?" > s$:s$ ??s$="acceleraton""No, I mean what's the equation":830> A (s$)7s$=s$(7) B3s$(11)=(r(1)*2)praise F(:"OK, so put t=2 into the equation": H"What's the value of dv/dt ?" J7exp=r(1)*4:egc=3:return=848P Kgetans Mscrhdr P$"Would you like another question?" R s$:s$ T$s$(11)="y"720 W$s$(11)="n"760 Z&"Please answer yes or no":848P \6start=800 :return=865a:n$="acceleration" ^endsec a     integration  :"Integration": "Integration is the opposite process to differentiation. In case you've forgotten how to differentiate, here's the rule again:": "If y=x then dy/dx=nx" :"In differentiating, the index islowered by one and the result multiplied by n. The opposite process - integration - is therefore to raise the index by one and to divide by n+1." scrhdr #"ie. if dy/dx=x then y=x/(n+1)" q:"another way of writing this is to say that the integral of x is x/(n+1), the shorthand for which is:": "xdx = x/(n+1)" :"Because constants ( terms not involving variables like x & y )disappear on differentiating, they reappear on integrating. The constant is unknown, and represented with a c.": scrhdr ."so the integration formula becomes:": "xdx = x/(n+1) + c" :"eg. 3xdx = x+c": " 8xdx = 4x+c": " 15xdx = 5x+c": qsthdr }"To get , use GRAPHICS H to get , use GRAPHICS I. Don't forget to go back out of graphics mode afterwards." -:"what is ";r(1)*3;"xdx ?":  a$:a$ /a$(11)=(r(1))9720% *a$(11)(r(1))"No." (:"would you like another question?":  a$:a$ $a$(11)="y"945 $a$(11)="n"970 "answer yes or no":958 scrhdr  what's it for ?  :"You might be wondering what integration is for. The simple answer is that it is used to find the area under a curve between two points. Why you would want to do this is anothermatter. Anyway, here's an example" scrhdr "To find the area under the curvey=3x between x=1 and x=0, you could plot it on graph paper and count the squares. This would take a long time, though, and would be only as accurate asyour estimates for the sizes of the fractions of squares." |"A better way is to integrate the equation for the curve, and evaluate it for the two boundaryvalues of x=0 and x=1.": scrhdr 1"This is what the graph of y=3x looks like:": 3,16;"x=1":4,2;"y":7,20;"3xdx=x+c":30,60<:0,80P:30,60<:120x,0 m130,60<:0,80P:30,61=:z=14:28,z^3/1.3&fff:z h::::::::"area=[x+c(where x=1) - x+c(where x=0)]"::"which is written [x]" scrhdr ."first evaluate x for x=1 x=1=1 next evaluate x for x=0 x=0=0 now subtract the second value from the first: area=1-0=1" scrhdr 6start=900:return=1010:n$="integration" endsec  K x$="": o=115 n=18 #w(o,n):x$(oo)+n-1,w(o,n) n o  H24,16,16,32 ,16,16,24,0 C24,8,8,4,8,8,24,0 F0,0,62>,64@,64@,64@,62>,0 E0,0,124|,2,2,2,124|,0 G0,66B,66B,66B,66B,66B,44,,0 G0,44,,66B,66B,66B,66B,66B,0  G0,0,60<,64@,124|,64@,60<,0 I240,8,120x,128,248,0,0,0 C0,0,0,4,126~,4,0,0 I248,136,136,136,0,0,0,0 C4,4,228,4,0,0,0,0 G240,8,240,8,240,0,0,0 G28,20,16,16,8,8,40(,568  E4,68D,228,68D,0,0,0,0 "G16,16,16,0,0,480,72H,480 4 #' #)::9200#+((*6)+1): #"Berk ! What's ";c$;" ?": #'"Wally ! ";c$;" isn't on the tape.": ##"Moron ! Try spelling straight": #"sorry, ";n$;" ? ": #"Are you tired, ";n$;" ?": #"Don't wind me up, ";n$: $T $V scrhdr() $Y $\9900&:: $ $ qsthdr() $ $'9600%:9900&:egc=0: $ $ $endsec(start,n$) $ $5"That finishes ";n$:"Hit r to repeat this section" $ 9900& $a$="r"start $return $ $ $getans(loop,exp) $ $?dp=0:err=0:u$:u$:(u$)>10 err=1 $(u$)=09460$ $[j=1̱(u$):(u$(jj))<45-ů(u$(jj))>579ů(u$(jj))=47/err=1 $%9480%:j:u$="-."err=1 $Cerr=1dp>1"type the answer as a number":9460$ $=ans=(u$):(exp-ans)>0.005y# ="No, the answer is ";exp $$(ans-exp)<0.005y# =9720% $return $ % %$(u$(jj))=46.dp=dp+1 % ,j>1Ư(u$(jj))=45-err=1 %  %k9800H&:d1=r(1)/10 +r(2)/100d+r(3)/1000:dec=d1:int=1000*dec %& %A"See if you understand this by trying the following examples" % 9800H& % %"True or false ?" %a$ %a$ %a$"t"a$"f"9650% % %n:n;"/";:d:d: % 9630% %h"No. Don't add the numbers together. It's just the second number, so the answer is ";r(2) % %K"No, its the second number, not the first, so the answer is ";r(2) % %%9720%+((*9 )+1): % "Good": % "Correct": %"Right again": %"OK": %"That's it": %"Brilliant !": %"Well done": &"Very good": &"Genius !": &HWz=110 :r(z)=(*9 )+1:z:r(1)r(2)9800H&: &R#r(1)r(2)9800H& &\ &z "trap": & &"Hit C to continue" &a$= &a$="c"9950& &a$="r"9950& & 9920& & &z=1502:z & % >@@@>||BBBBB,,BBBBB<@|@<x~(8DD0H0X 5 crhdT$sthd$ndse$etan$iagrarais%!