Welcome! [smartfuse.s3.amazonaws.com]smartfuse.s3.amazonaws.com/2edcdb4fae2b1d953afcf227a969048e/... · Welcome! This is the first in a series of teaching aids designed by teachers

Level 7/8 Pack 1. Page 1. [email protected]

Welcome!

This is the first in a series of teaching aids designed by teachers for teachers at level 7/8.The worksheets are designed to support the delivery of the National Curriculum in a variety ofteaching and learning styles. They are not designed to take the pedagogy away from the teacher.The worksheets are centred around the shown level, but spiral from the level below to the levelabove. Consult the National Numeracy Strategy for definitive National Curriculum levels.They can be used by parents with the support of the on-line help facility at www.10ticks.co.uk.

Contents and Teacher Notes.Pages 3/4. Probability 1.

Revision of level 5 and level 6 single event probability, including theprobability of events not happening.

Pages 5/6. Probability 2.Mutually exclusive and exhaustive events. Pupils have to be familiar withfraction addition and subtraction as well as with decimal addition andsubtraction. The addition (OR) rule.

Pages 7/8. Experimental Probabilities 1.Assigning probabilities. Finding the expectation of a probability.

Page 9. Experimental Probabilities 2.Experiments to compare theory and experiment. Pupils will use theexperimental probability diagrams (p 10) to find out experimental probabilitiesand compare them with theoretical probabilities.

Page 10. Experimental Probability Diagram.A grid for recording experimental probability for an event. After 50 trials readoff the experimental probability. This could be used for some of the level 5probability. Photocopy this sheet back to back so pupils will only need onesheet to complete Experimental Probabilities 2.

Pages 11/12. Experimental Probabilities 3.Relative frequency and relative frequency graphs. A pictorial way to showrelative frequency "settling down" to theoretical probability with the greater thenumber of trials. "140 trials" has been chosen so as to fit on 1 piece of A4 graphpaper landscape (long ways).

Pages 13/14. Lists and Possibility Spaces.Revision of the Level 6 descriptors.

Pages 15/16. Tree Diagrams (Independent Events).Complete the tree diagram by adding the appropriate probabilities and thensolve the questions. Drawing tree diagrams from worded questions.

Pages 17/18. 'AND' and 'OR' Rules.The multiplication rule (AND rule). Questions that involve the AND\ OR rules.

Pages 19/20. Substitution (Negative Numbers and Fractions).Before pupils can plot linear functions they must be comfortable with this typeof substitution, with and without a calculator. The formulae chosen are typicalof the formulae they will meet around the school.

Pages 21/22. Plotting Linear Functions.Revision of all the level 6 skills. Finding the gradient and y intercept of a line.Rearranging linnear equations. Finding the coordinate of the mid-point of aline segment.

Page 23. Finding the Equation of a Straight Line.Given two coordinates, pupils have to find the equation of the line. Scales arechanged on the axes so that pupils realise they have to read off axes and notcount squares.

[email protected] 7/8 Pack 1. Page 2.

Page 24. Finding Equations of Straight Line Graphs 1.The lines are drawn already, pupils have to find the equations.

Pages 25/26. Finding Equations of Straight Line Graphs 2.As above.

Pages 27-30. Linear Function Snap Cards.These cards can be used in a variety of ways. They may be used as a simplepairing exercise to stick in the book. They may be used in a game of "pairs",cards are turned upside down, two are turned over. If they match you win them.Use 2 sets of each sheet and play "snap!" The list is as endless.

Pages 31/32. Scatter Graphs 1.Plotting scatter graphs and finding the line of best fit by inspection. The line ofbest fit is then used to find some estimations.

Pages 33/34. Scatter Graphs 2.Scatter graphs are extended by finding the mean of the scores which is plottedto aid positioning the line of best fit. Then the equation of lines of best fit haveto be found. This equation is used to find further estimates of values.

Pages 35/36. Direct Proportion.Finding the constant of proportionality to solve direct proportion questions.

Pages 37/38. Quadratic Sequences.Revision of finding the nth term of linear sequences. Simple and more difficultquadratic sequences.

Pages 39/40. Finding Quadratic Functions (Difference Method).Only required for coursework. This is a useful sheet to do the week beforeattempting coursework with a class. It shows pupils how to derive the formulafor a quadratic using the difference method. This enables them to score highermarks in strand iii at AT 1.Note: this is not a justification of a formula.

Pages 41/42. Practical Quadratic Number Patterns.Looking at practical patterns that derive quadratic functions. These are in thestyle of "Matchsticks" investigations pupils may meet through coursework.

Pages 43/44. Investigations (Quadratic Patterns).Some investigations that again derive quadratic functions. The first set isgrouped together through a football theme, each one is an investigation so alldon't have to be completed. The onions investigation is a very good one tostretch the more able and keep the rest going! It is also quite simple to justifythe quadratic (see above). Regroup the onions into squares, each diagram willmake 2 squares n2 and (n + 1)2 in dimension. By working out the bracket andadding them all together it is fairly simple to derive 2n2 + 2n + 1.For the very able, the first pattern can be extended into 3 dimensions leading tothe cubic 4/

3 n3 + 2n2 + 8/

3n + 1 !!!

Copyright in Worksheets. ©Fisher Educational Ltd. 2000.Copyright in the worksheets belongs to Fisher Educational Ltd. Each purchase of the worksheets represents alicence to use and reproduce the worksheets as set out in the Terms and conditions shown on the 10ticks website.

'10TICKS', and '10TICKS.co.uk' and/or other 10TICKS services referenced on this web site or on the Worksheetsare trademarks of Fisher Educational Ltd in the UK and/or other countries.

Details of copyright ownership in the clip art used in these worksheets:Copyright in the clip art used entirely in this pack is owned by Nova Development Corporation, California, USA.


Probability 1.Single Event (Revision).

P (event) = Number of ways the event can occur Total number of outcomes

1). The following spinners are spun.What is the probability of landing on the shaded section for each spinner ?a). b). c). d). e).

2). Counters are placed in a box. For each of the following boxes, find the probability of ashaded counter being drawn out of the box at random.a). b). c). d). e).

3). If a fair six-sided dice is thrown, what is the probability that the score is :-a). a 5 (five), b). an odd number, c). less than 4,d). a prime number, e). a square number, f). greater than 4 ?

4). There are 12 counters in a box numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. If onecounter is drawn out at random, what is the probability that it is a counter :-

a). with an odd number, b). greater than 4,c). with a square number, d). that is greater than or equal to 8 ?

5). I have a pack of playing cards, containing 52 cards. I pick a card at random. What is theprobability that the card I select is :-

a). a queen, b). a red card, c). a club,d). an ace, e). a black ace, f). the jack of diamonds ?

6). Writing pads are made in four different colours. In a box there are 3 blue, 10 red, 6 whiteand 5 green pads. What is the probability when I open the box I randomly pick a :-

a). blue pad, b). red pad, c). white pad,d). green pad, e). yellow pad ?

7). Draw the number line in your book.The number line represents all possibleprobabilities.

Indicate, by arrows labelled "a" to "f" the probabilities for :-a). throwing a HEAD on a coin, b). cutting a spade from a pack of cards,c). October following September, d). you visiting the moon this evening,e). scoring a 4 on a die, f). your teacher winning the lottery.

8). Arrange these events in order of which is "most likely" down to which is "least likely":-a). throwing a dice and getting a "6",b). being born on a day with "y" in it,c). spinning a coin and it landing "heads",d). somebody having their birthday on Christmas day.

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P (event not happening) = 1 - P (event happening).

E.g. In a bag are 7 marbles, 4 are red. One is drawn at random.What is the probability that the marble drawn is not red ?

P (not red) = 1 - P (red) P (not red) = 1 - 4/7

P (not red) = 3/7

9). A biased coin has a probability of landing on heads of 0.7. What is the probability ofthe coin landing on tails ?

10). In a bag there are 10 marbles. Four are red. One marble is drawn at random.What is the probability of picking a marble that is not red out of the bag ?

11). In a cake shop the probability of a customer buying a cream cake is 5/8.

What is the probability of a customer not buying a cream cake ?

12). Another biased coin has a probability of landing on tails of 0.32.What is the probability of landing on heads ?

13). A fair die is rolled. What is the probability of the die landing on a number that isn't 6 ?

14). Year 7 and Year 10 boys play a football match with a penalty shoot-out, so that the teamscannot draw. The probability of Year 7 winning is 0.28. What is the probability ofYear 10 boys winning ?

15). Year 8 and Year 9 girls play a netball match with a penalty shoot-out, so that the teamscannot draw. The probability of Year 9 winning is 0.75, what is the probability of Year 8winning ?

16). Ted plays a game of chess against a computer, it cannot draw. The computer is set tohave a 0.63 chance of winning the game. What is the probability that Ted wins the game ?

17). Alex plays a game of patience on the computer. The computer is programmed to letthe player have a 4/

9 chance of winning. What is the probability that Alex loses ?

18). A bag contains 5 blue discs, 6 white discs, 9 green discs and 10 red discs. One is drawn outat random. What is the probability of not picking :-a). a white disc, b). a green disc, c). a blue disc, d). a red disc ?

19). Draw out the number line in your book.The number line represents all possibleprobabilities.

Indicate, by arrows labelled "a" to "f" the probabilities for :-a). being born on a week day, b). not cutting a club from a pack of cards,c). Tuesday following Monday, d). your teacher laying an egg,e). not scoring a 6 on a dice, f). the next baby born is a girl.

20). Arrange these events in order of which is "most likely" down to which is "least likely":-a). throwing a dice and not getting a 2,b). being born on a day with an "s" in it,c). not cutting a king from a pack of cards,d). not being born on a week day.

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Probability 2.

Mutually Exclusive and Exhaustive Events.

When events cannot happen at the same time they are called mutually exclusive.

E.g. If a coin is thrown P (landing on Heads) = 1/2 and P (landing on Tails) = 1/

2 .

It is impossible for the coin to land on Heads and Tails at the same time.These are mutually exclusive.In this example there are no other possible outcomes and these are also called exhaustive events.

The probabilities of exhaustive events add up to 1.

1). At a zebra crossing you either wait or walk. The probability at a particular zebra crossingto wait is 0.17. What is the probability that you will be able to walk ?

2). John's favourite drawing pin can land point down or point up. The probability for thedrawing pin to land point down is 0.42. What is the probability it lands point up ?

3). The Year 8 football team can win, draw or lose. The probability they win is 0.5, theprobability they draw is 0.2. What is the probability they lose ?

4). In the school canteen Jenny can choose from chips, pasta or baked potatoes.The probability she chooses chips is 5/

8, the probability she chooses pasta is 1/

8.

What is the probability she chooses baked potatoes ?

5). Billy the cat eats out of a red bowl or a yellow bowl or a green bowl. The probability heeats out of the red bowl is 0.53 and the probability he eats out of the green bowl is 0.39.What is the probability he eats out of the yellow bowl ?

6). Year 10 hockey team can win, draw or lose. The probability they win is 3/10

, theprobability they draw is 1/

5. What is the probability they lose ?

7). A shop sells three choices of sandwiches, chicken, egg and salad. The probability acustomer chooses chicken is 1/

3. The probability a customer chooses egg is 1/

4.

What is the probability that a customer chooses salad ?

8). A motorist has the choice of 3 car parks in town A, B and C. The probabilityhe parks in Car Park A is 0.46, the probability he parks in Car Park B is 0.17.What is the probability he parks in Car Park C ?

9). At a junction in the road there are 3 choices which a motorist can take, route 1, 2, or 3.The probability a motorist takes route 1 is 5/

9 , the probability a motorist takes route 2 is 1/

4.

What is the probability a motorist takes route 3 ?

10). A four sided spinner is cut out of card. It is not fair. It is labelled 1, 2, 3 and 4.The probability it lands on side 1, is 0.4, the probability it lands on side 2 is 0.17, theprobability it lands on side 3 is 0.24. What is the probability it lands on side 4 ?


Addition Rule (OR Rule)

When two events are mutually exclusive we can work out the probability ofeither of them occurring by adding together both the individual probabilities.

E.g. A bag contains 5 red marbles, 3 yellow marbles and 2 green marbles. One is picked at random.What is the probability it is a red or yellow marble ?

P (red marble or yellow marble) = P (red marble) + P (yellow marble )= 5/

10 + 3/

10= 8/

10 = 4/

5 .

1). Traffic lights can show red, red/amber, amber or green. The probability of showingamber is 0.17, the probability of showing red/amber is 0.14.What is the probability of showing amber or red/amber ?

2). In a flower contest the probability that a red rose will win is 0.18, the probability that ayellow rose will win is 0.24. What is the probability that a red or yellow rose will win ?

3). A fair die is rolled. What is probability it landing on a 3 or a 4 ?

4). There are 4 doors in an office. The probability that Janet enters by door 1 is 0.06 and theprobability she enters by door 2 is 0.66. What is the probability she enters by either door 1 or 2 ?

5). A bag contains 8 blue discs, 12 orange discs and 4 red discs. If a disc is picked at randomwhat is the probability of getting :-a). a blue disc, b). an orange disc, c). a blue or red disc,d). a blue or orange disc, e). not an orange disc, f). a blue, orange or red disc ?

6). A small box of chocolates contains 3 hard centres, 8 soft centres and 7 chewy centres.What is the probability of picking :-a). a hard centre, b). a hard or soft centre, c). a soft or chewy centre,d). a hard or chewy centre,e). not a soft centre, f). not a soft or hard centre ?

7). In a class of thirty pupils 9 play hockey, 12 play football, 5 play rugby and 4 goswimming. If a pupil is selected at random, what is the probability that the pupil willa). play football, b). play hockey or swim, c). play hockey or football,d). not play rugby, e). not swim, f). not play rugby or swim ?

8). In a cash bag there are six 1 pence coins, eight 2 pence coins, twelve 5 pence coinsand four 10 pence coin. If a coin is drawn at random, what is the probability that the coin :-a). is a 5p, b). is a copper coin, c). is not a 10p,d). is a 1p or a 5p, e). is not a copper coin, f). is not a 1p, 2p or 5p?

9). A box contains 12 blue discs, 10 green discs, 3 yellow discs and 15 red discs. If a disc ispicked at random what is the probability of getting :-a). a blue disc, b). a green disc, c). a blue or yellow disc,d). a blue or green disc, e). not a yellow disc, f). a pink disc ?

10). A box contains 4 blue discs, 6 green discs, 18 purple discs and 12 red discs. If a disc ispicked at random what is the probability of getting :-a). a red disc, b). a green disc, c). a blue or red disc,d). a purple or green disc, e). not a purple disc, f). a blue, green, purple or red disc ?


Experimental Probabilities 1.

Assigning Probabilities.Previously we have seen 3 ways of assigning probabilities.

A -- Use equally likely outcomes.B -- Look back at data.C -- Use a survey or experiment to collect data.

Which of these methods would you use to find :-

1). The probability a volcano will erupt next year in a particular country.2). The probability a person chosen at random in a school will be right handed.3). The probability a biased coin will land on Tails when thrown.4). The probability a boy's name will be picked at random out of 30 girls and 30 boys.5). The probability that a car is stolen on a Friday night in Manchester.6). The probability a drawing pin will land point up when dropped.7). The probability a fair die will land on 2 when rolled.8). The probability that I win a raffle if I buy 50 out of the 200 tickets on sale.9). The probability that it will rain on Easter day.10). The probability that the king of clubs will be chosen from a pack of cards.

Using Survey or Experiment.Using our surveys or experiments we can determine a probability for an event. From theseprobabilities we can predict how many times we would expect a particular event to occur for acertain number of trials. This is the expectation, not what will happen. (The expectation shouldbe a very close model as to what will happen! ). To find the expected number of outcomesmultiply the probability of the event by the number of trials.

E.g. A police car stops 100 cars at random. 15 drivers did not have road tax.a). What is the probability that a driver doesn't have road tax ?

15 = 3100 20

b). If the police car stops another 360 cars, how many might they expectto have no road tax ?

15 x 360 = 54100

1). A coin is to be thrown in an experiment. How many times would you expect it to land onheads if it is thrown :-a). 200 times, b). 900 times, c). 2450 times, d). 9446 times ?

2). A fair die is to be rolled in another experiment. How many times would you expect the dieto land on 6 if it is rolled :-a). 120 times, b). 600 times, c). 792 times, d). 21246 times ?

3). If you carried out the experiments in questions 1). and 2). would you expect toget exactly these results ?


4). Here are the results of a survey of cars passing the school.

Colour Red Blue Green Yellow OtherNumber of cars 10 15 8 5 2

a). What is the probability of the next car passing the school being :-i). red, ii). green, iii). yellow, iv). not blue ?

b). If 800 cars pass the school, how many would you expect to be red ?c). If 320 cars pass the school, how many would you expect to be yellow ?d). If 120 cars pass the school, how many would you expect to be blue ?

5). A drawing pin is repeatedly dropped in an experiment to see which way up it will land.Here are the results.

Outcome FrequencyPoint up 180Point down 120

a). What is the probability of the drawing pin landing :-i). point up, ii). point down ?

b). If the drawing pin is dropped 800 times how many times would you expect it to landpoint up ?

c). If the drawing pin is dropped 245 times how many times would you expect it to landpoint down ?

6). Jean conducts a survey on pupils favourite snacks. Here are the results:-

Snack Crisps Chocolate Fruit Biscuit OtherFrequency 11 10 12 15 2

a). What is the probability of a pupil :-i). liking Crisps, ii). liking Fruit, iii). liking Biscuits, iv). not preferring chocolate ?b). If 800 pupils were asked, how many would you expect to prefer biscuits ?c). If 150 pupils were asked, how many would you expect to prefer crisps ?d). If 470 pupils were asked, how many would you expect to prefer chocolate ?

7). In an experiment, 100 seeds are sown, but only 45 germinate.a). What is the experimental probability of a seed :-

i). germinating, ii). not germinating ?b). If 40 seeds are sown, what number might be expected to germinate ?c). If 1340 seeds are sown, what number might be expected not to germinate ?

8). 80 trains arrived at Lostock Station this morning. 14 arrived early, 6 were late and the restwere on time.a). What is the probability that the next train will be:-

i). late, ii). early, iii). on time ?b). 120 trains arrive in the afternoon, how many might you expect to be early ?c). Tomorrow 360 trains are due, how many would you expect to be late ?d). This week 2876 trains should come through Lostock Station,

how many would you expect to be on time ?



Theoretical and Experimental Probabilities.

You will need an Experimental Probability Diagram and 4 coloured pencilsfor each question below.

1). You are going to find the experimental probability for throwing a coin and it landing onHeads.On the Experimental Probability Diagram under Event 1 write in "Not a Head (Tails)",under Event 2 write in "Heads".Throw a coin 50 times and record each outcome on the Experimental Probability Diagram.Every time you throw a Head you move to the right, every time you throw a Tail you moveto the left. Read off the probability when you reach the bottom and record it.Repeat this experiment again another 3 times. Record the outcomes with a different coloureach time. Read off the probability for each set of throws.a). What is the theoretical probability of throwing a coin and it landing on Heads ?b). Do all 4 sets of throws agree exactly with the theory ?c). Do all 4 sets of throws get exactly the same experimental probability ?d). Find the mean of the 4 experimental probabilities.e). Is the mean of the 4 experimental probabilities close to the theoretical probability ?

2). You are going to find the experimental probability for rolling a die and it landing on a 6.On the Experimental Probability Diagram under Event 1 write in "Not a 6", under Event 2write in "6".Roll the die 50 times and record each outcome on the Experimental Probability Diagram.Every time you roll a 6 you move to the right, every time you roll another number youmove to the left. Read off the probability when you reach the bottom and record it.Repeat this experiment again another 3 times. Record the outcomes with a different coloureach time. Read off the probability for each set of throws.a). What is the theoretical probability (as a decimal) of rolling a 6 on a die ?b). Do all 4 sets of rolls agree exactly with the theory ?c). Do all 4 sets of rolls get exactly the same experimental probability ?d). Find the mean of the 4 experimental probabilities.e). Is the mean of the 4 experimental probabilities close to the theoretical probability ?

3). Find the experimental probability of dropping a drawing pin and it landing point up.On the Experimental Probability Diagram under Event 1 write in "Point Down", underEvent 2 write in "Point Up".Drop the drawing pin 50 times and record it on the Experimental Probability Diagram.Every time the drawing pin lands point up you move to the right, every time it lands pointdown you move to the left. Read off the probability at the bottom and record it.Repeat this experiment again another 3 times. Record the outcomes with a different coloureach time. Read off the probability for each set of throws.Find the mean of the 4 experimental probabilities.

4). Make up your own experiment. Decide which event you are trying to find an experimentalprobability for. This will be Event 2. Use the Experimental Probability Diagram to helpfind an accurate experimental probability for the event.


Experimental Probability Diagrams

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70.

60.

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40.

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Eve

nt 2

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Comparing Theory to Experiments.If we know the theoretical probability of an event we can compare it to the experimentalprobability (relative frequency) by setting up a number of trials. By drawing a relative frequencygraph, the experimental and theoretical probabilities can be compared over all the trials.

The relative frequency is the experimental probability after a given number of trials.

or relative frequency of an event = the number of times the event occurs the number of trials

Experiment 1. Throw a coin 140 times and record the number of times the coin lands on tails.a). Copy the table below.

No. of throws No. of times landed Relative Frequency Relative Frequency (Trials, t) on tails so far (n ). ( P (Tails) = n ÷ t ) 2 decimal places

1 2 3 4 5 10 15

140

b). In the table we need to work out the relative frequency for the first 5 trials, after thatevery 5th trial, i.e. the 10th, 15th, 20th, 25th, .... This will make plotting the relativefrequency graph much easier.Now throw the coin and fill in the table after each trial.

c). Draw the axes for the relative frequency graph like those below.d). Draw the theoretical probability line of the coin landing "tails".e). Plot the relative frequency from the table.

f). What do you notice about the relative frequency and the theory line as the number oftrials increases ?

Theory line

No. of trials

0.50

1.00

0

Rel

ativ

e Fr

eque

ncy


No. of trials

0.50

1.00

0

Rel

ativ

e Fr

eque

ncy

Theory line

Experiment 2. Roll a die 140 times and record the number of times the die lands on 6.a). Copy the table below.

No. of throws No. of times landed Relative Frequency Relative Frequency (Trials, t) on 6 so far (n ). ( P (6) = n ÷ t ) 2 decimal places

1 2 3 4 5 10 15

140

b). As before, work out the relative frequency for the first 5 trials, after that every5th trial, i.e. the 10th, 15th, 20th, 25th, ....Now roll the die and fill in the table after each trial.

c). Draw the axes for the relative frequency graph like those below.d). Draw the theoretical probability line of the die landing on 6.e). Plot the relative frequency from the table.

f). What do you notice about the relative frequency and the theory line as the number oftrials increases ?

1). Make up your own probability experiment. Choose an event that you already know thetheoretical probability for. Record the relative frequency in a table and then plot a relativefrequency graph for it.What do you notice about the relative frequency and the theory line ?

In experimental probability terms 200 trials are insignificant. These experiments should havethousands, if not tens of thousands of trials to become significant.

The more times you do an experiment the closer the relative frequency will get to the theory.

2). Choose an event you don't know the theoretical probability for.Here are some ideas : Which way will a shoe land when pushed off a table ?

Which way up will a drawing pin land when dropped ?Which way will buttered toast land when falling off a plate ?

Record the relative frequency in a table and then plot a relative frequency graph for it.Assign a probability to your event.


Lists and Possibility Spaces.

1). Katherine goes to the canteen. She wants a hot and a cold drink with her meal.Write all the permutations she can choose from the list of options below.

Hot Drinks: Tea Coffee Hot ChocolateCold Drinks: Cola Lemonade Milk

2). On an activity holiday guests have to choose an indoor and an outdoor sport on the first day.Write all the permutations a guest can choose from the list of options below.

Indoor: Darts Pool Snooker Table-tennisOutdoor: Sailing Walking Climbing

3). Sabrina wants to buy a television and video.a). Write all the permutations she can choose from the list of makers below.

Television: JVC Sony Toshiba AkaiVideo: Samsung Sony Panasonic Akai

b). If she chooses each at random what is the probability she chooses :-i). both Sony,ii). different makers,iii). the same makers ?

4). Sanjid and Arthur go to the drinks machine. They buy a drink each.The drinks machine has Coffee, Tea and Hot Chocolate.a). List all the permutations of drinks they could buy.b). It is equally likely as to which drinks they buy, find the probability that they :-

i). both get tea,ii). buy one coffee and one tea,iii). buy at least one Hot Chocolate.

5). James has a circular and a triangular spinner numbered as shown.On each spinner it is equally likely to land on any of thenumbers. James spins them both.a). List all the possible outcomes he could get.b). Draw a possibility space for the same events.c). He multiplies the two scores together. Find the probability he gets a :-

i). score less than 20,ii). score greater than 15,iii). square number.

6). A fair six sided die and a coin are thrown together.a). List all the different permutations of how the die and coin could land.b). Draw a possibility space for the same events.

Using either find the probability of getting :-i). the probability of a Tail and a 6,ii). the probability of a Head and a 2,iii). the probability of a Tail and a 7.

2 5

351

7


7). A fair spinner has four numbers 2, 2, 3, 5. It is spun twice. The sum of the scores is noted.Draw a possibility space and find the probability that :-a). the sum is 4,b). the sum is 6 or more,c). the sum is 9,d). the sum is a square number.

8). a). Two fair coins are thrown.i). List all the different permutations showing how the coins could land.ii). Draw a possibility space for the same events.Using either find the probability of getting :-iii). two Heads,iv). one of each.

b). Three fair coins are thrown.i). List all the different permutations showing how the coins could land.ii). Can you draw a possibility space for the three coins ?Find the probability of getting :-iii). three Heads,iv). two Tails and one Heads,v). at least one Head.

9). Two fair six sided dice are thrown and the difference between the dice noted.Draw the possibility space. What is the probability that :-a). there is no difference between the two scores,b). there is a difference of 3 between the two scores,c). there is a difference of more than 4 between the scores ?

10). In a game a normal fair dice is rolled, then a card is picked at random from 5 cardsnumbered from 1 to 5. Draw a possibility space. Find the probabilities that :-a). the numbers are both 5,b). the sum of the numbers is 8,c). the numbers are both the same,d). the sum of the numbers is 4 or 5.

11). Two unbiased dice are numbered 3, 3, 4, 4, 6, 8. They are thrown together and thetotal of the two scores is found. Draw a possibility space. Find the probabilities that :-a). the numbers are both the same,b). the sum is 9,c). the sum is an even number,d). the sum is greater than 7.

12). A three sided spinner is numbered 1, 4, 5. A five sided spinner is numbered 2, 3, 4, 6, 8.Both are spun at the same time. Draw a possibility space and find the probability that :-a). the sum of the scores is a square number,b). the product of the scores is greater than 20,c). the difference in the score was exactly 1.

13). There are 2 bags of marbles. The first contains 3 red, 1 blue and 3 green, the secondcontains 1 red, 2 blue and 2 green. A marble from each is removed, draw the samplespace. Find the probability of getting :-a). 2 red marbles, b). 2 blue marbles, c). a red and blue marble,d). 2 green marbles, e) a green and blue marble.


Tree Diagrams (Independent Events).

1). Each morning Bob and Bill catch the same bus. Theprobability that Bob catches the bus is 0.9 and for Billit is 0.7. The probabilities are independent of each other.a). Copy and complete the tree diagram.b). Calculate the probability that on a given day :-

i). they both catch the bus,ii). Bob catches the bus, but not Bill,iii). neither catch the bus,iv). at least one of them catch the bus.

2). There are 10 books on a shelf in a library. Seven are fictionand three are nonfiction. A member of the public takes abook at random, looks at it, and then replaces it on the shelf.Another member of the public then takes a book at randomfrom the shelf.a). Copy and complete the tree diagram.b). What is the probability the two books taken are :-

i). both nonfiction,ii). both fiction,iii). one of each ?

3). The Post Office have stated that 94% of all firstclass letters are delivered the next day.Ron posts 2 letters on Friday.a). Copy and complete the tree diagram.b). What is the probability that :-

i). both letters are delivered on Saturday,ii). neither letter is delivered on Saturday,iii). just one of the letters arrives on Saturday,iv). at least one letter is delivered on Saturday ?

4). In a box there are 6 red and 4 green counters, all of thesame size. One is drawn and then replaced. A secondcounter is then drawn.a). Copy and complete the tree diagram.b). What is the probability that :-

i). both counters are red,ii). both counters are green,iii). one is red and one green,iv). at least one is green,v). neither is green ?

5). A fair coin is thrown three times. Copyand complete the tree diagram, markingthe probabilities and outcomes. What isthe probability that :-a). all three coins are Heads,b). exactly two coins land on Tails,c). at least one coin lands on Tails,d). no coins land on Heads ?

First draw

Second draw

Red

Green

3 5

2 5

Red

Red

Green

Green

First throw

Second throw

Tails

Third throw

Heads

Heads

Heads

Tails

Tails

Tails

Tails

Tails

Heads

Heads

Heads

Heads

Tails

Bob Bill

0.9 catch

not catch

catch

catch

not catch

not catch

0.7

First book

Second book

fiction

nonfiction

710

fiction

fiction

nonfiction

nonfiction

310

First letter

Second letter

ArriveSaturday

0.94

Not ArriveSaturday

ArriveSaturday

ArriveSaturday

Not ArriveSaturday

Not ArriveSaturday

0.94


Saturday Sunday

Sunny0.2

Cloudy

Rainy

0.5

Sunny

Sunny

Sunny

Cloudy

Cloudy

Cloudy

Rainy

Rainy

Rainy

6). On any particular day in Bolton it can be sunny,cloudy or rainy. The probability of it being sunnyis 0.2, cloudy 0.3 and rainy 0.5. Copy and completethe tree diagram for this weekend.Find the probability that :-

a). both days are sunny,b). both days are rainy,c). one day is sunny and one day it rains,d). it does not rain at all.

7). Four fifths of cars on roads today are of foreign manufacture. A pupil looks out of thewindow during Science and watches two cars go by. Draw a probability tree for thisshowing all the probabilities and outcomes.Hence find the probability that :-

a). both cars are British, b). both cars are foreign,c). there is one of each, d). at least one car is not British.

8). When Alex goes to school, she either goes by car (probability 0.3) or catches the bus(probability 0.7). When she comes home from school she either goes by car (probability0.2), catches the bus (probability 0.5) or walks (probability 0.3). Draw a probabilitytree for going to and from school, and show all the probabilities and outcomes.Hence find the probability that :-a). she goes and comes back by car, b). both journeys are on the bus,c). at least one of her journeys is by bus, d). she will travel in a vehicle for both,e). at least one of her journeys is by car, f). she walks to and from school.

9). In a bag are 5 red, 3 blue and 4 green counters of equal size. One is picked, the colournoted and then put back into the bag. A second is then drawn. Draw a probability tree forthis and show all the probabilities and outcomes.Hence find the probability that :-

a). both counters are red, b). one red and one green are drawn,c). at least one blue is picked, d). at least one red is picked,e). neither are green.

10). A couple have three children. It is equally likely that each child is a boy or a girl. Drawa probability tree for this and mark the probabilities and outcomes.Hence find the probability that :-

a). all three are boys, b). all three are girls,c). they have 2 girls and a boy, d). they have at least one boy,e). they have no girls.

11). A dice is rolled three times and it is noted whether a six is scored or not. Draw aprobability tree for this and mark the probabilities and outcomes.Hence find the probability that :-

a). all three are sixes, b). exactly one six is scored,c). at least two sixes are scored, d). at least one six is scored,e). no sixes scored.

12). On a fruit machine there are three drums, each with cherries, lemons and pears. Theprobability of a cherry is 0.5, the probability of a lemon is 0.1 and the probability of a pearis 0.4. Draw a probability tree for this and mark the probabilities and outcomes.Hence find the probability that :-

a). all are cherries, b). all are lemons,c). at least one lemon is picked, d). at least one cherry and pear is picked,e). none are pears.


'AND' and 'OR' Rules.

Two events are said to be independent if the outcome of one event does not effect the outcomeof the other event. The combined events we have looked at so far are independent events.

Multiplication Rule (AND Rule).

It is possible to calculate probabilities of combined events without using a tree diagram. Thiswould be the same as multiplying along the branches of a tree diagram.

Where there are two independent events A and B, the probability of A and Boccurring is the probability of A multiplied by the probability of B.

P(A and B) = P(A) x P(B)

N.B. The rule is similar if there are 3 or more events.

1). Jane goes shopping. The probability she buys a CD is 0.4.The probability she buys a magazine is 0.6. The events are independent.What is the probability she buys a CD and a magazine ?

2). In a football match the probability that Michael scores is 0.4.The probability that James scores is 0.2. The events are independent.What is the probability that Michael and James score in the football match ?

3). A coin and a normal fair die are thrown. Calculate the probability of getting :-a). a Tail and a five,b). a Head and a two.

4). Two fair dice are rolled. Calculate the probability of rolling a double six (6 and 6).5). For each of the first 4 questions draw a number line from 0 - 1. Mark on the number line

the position of P(A), P(B) and P(A and B) where A and B are the events.a). What do you notice about the position of P(A and B) ?b). Will there be any time when P(A and B) isn't the smallest probability ?

6). A playing card is drawn from the pack of 52 cards. It is replaced, the pack shuffled andanother card selected. Find the probability that :-a). the two cards were both red cards,b). the two cards were both diamonds,c). the two cards were both kings,d). the two cards were both black aces,e). a red card then a king were chosen.

7). A bag contains 3 red balls, 5 yellow balls and 2 green balls.A ball is selected at random, the colour noted then it is put back into the bag.A second ball is then selected at random from the bag.Find the probability that :-a). the two balls selected were both red balls,b). the two balls selected were both green balls,c). the first ball was yellow and the second ball was green.d). A third ball is selected. Find the probability all three balls selected are yellow.

8). A coin is thrown twice and the results noted. Calculate the probability of :-a). the coin landing on Heads both times,b). the coin landing on a Tail then on a Head,c). the coin landing on a Head and a Tail.d). What is the difference between question b) and c). ?The coin is thrown for a third time.e). What is the probability it lands on Heads for all three throws ?


AND and OR Rule.

Where the order of the outcome is not specified then we also need to use the OR rule.

E.g. Two fair dice are rolled. Calculate the probability of the dice landing on a 5 and a 6.

P(5 and 6 in any order) = P(5 and 6) or P(6 and 5)= (1/

6 x 1/

6) + (1/

6 x 1/

6)

= 1/36

+ 1/36

= 1/18

(Check this on a tree diagram).

1). A die is rolled twice. Calculate the probability of the die :-a). landing on an even number then a 5,b). landing on an even number and a 5,c). landing on a square number and a 2.

2). The letters of the word MISSISSIPPI are written on individual cards and then placed in abag. Each card is equally likely to be picked. A card is selected at random from the bag,placed back in the bag and a second card picked. Calculate the probability that :-a). the cards picked were both the letter S,b). the cards picked were both the letter P,c). the cards picked were a P then an S,d). the cards picked were an I and an S.

3). A playing card is drawn from the pack of 52 cards. It is replaced, the pack shuffled andanother card selected. Find the probability that :-a). the two cards were both clubs,b). the two cards were both red Kings,c). the first card was a two and the second card was red,d). one card was a Jack and the other card was red,e). one card was a club and the other card was a Queen.

4). A bag contains 4 red balls, 5 yellow balls and 3 green balls.A ball is selected at random, the colour noted then it is put back into the bag.A second ball is then selected at random from the bag. Find the probability that :-a). the two balls selected were both green balls,b). the first ball was red then the second ball was yellow,c). the two balls selected were a red and a green ball,d). the two balls selected were a yellow and a red ball.

5). The probability Jim is late for work on Monday is 0.3. The probability he is late for workon Tuesday is 0.2. Find the probability that :-a). Jim is late on Monday and Tuesday,b). Jim is on time on both Monday and Tuesday,c). Jim is late on only one of the days.

6). Emma sits her maths exam. The probability of her passing Paper I is 0.6 and theprobability of her passing Paper II is 0.8. Find the probability that :-a). Emma passes both her Paper I and Paper II,b). Emma fails both her Paper I and Paper II,c). Emma passes only one of her Papers.

7). At a golf course the probability of scoring a 'hole in one' at the first hole is 1/15

.The probability of scoring a 'hole in one' at the second hole is 1/

10.

Find the probability that for these first two holes :-a). a hole in one is scored on both holes.b). a hole in one is scored on the first, but not the second hole,c). a hole in one is scored on only one of the holes.


Substitution (Negative Numbers and Fractions).

1). Using the formula v = u + at, find the value of v ifi). u = 5, a = -8, t = 1 ii). u = 12, a = -8.4, t = 2 iii). u = 20, a = -4.6, t = 2 1

2 5 4iv). u = 12, a = -7.3, t = 6 v). u = 4, a = -0.3, t = -2 vi). u = 7, a = -3.1, t = 3 3

7 5 5

2). Using the formula y = mx + c, find the value of y ifi). c = -3, x = 4, m = 1 ii). c = -3, x = 5, m = 2 iii). c = 8, x = -7, m = 2 1

2 3 2iv). c = 4, x = -2.4, m = 3 v). c = -2, x = -3.6, m = 1 vi). c = -4.5, x = -4, m = 1 3

4 4 4vii). c = 3, x = -7, m = -2 viii). c = -4, x = -1.2, m = -2 ix). c = -2, x = -5.1, m = -4 2

3 5 3

3). Using the formula a = v2 - v

1 , find the value of a (acceleration) if

ti). v

1 = 15, v

2 = 8, t = 1 ii). v

1 = 1.4, v

2 = 0.8, t = 2 iii). v

1 = 10, v

2 = 3.5, t = 3 1

2 5 5

4). Using the formula s = ut + 1 at2, find the value of s if 2

i). u = 5, a = -8, t = 1 ii). u = 12, a = -3, t = 8 iii). u = 10, a = -3.6, t = 2 1 2 5 4

iv). u = 12, a = -2, t = 2.4 v). u = 6, a = -0.3, t = 2 vi). u = 7, a = -3.1, t = 5 15 5 5

5). Using the formula v2 = u2 - 2as, find the value of v ifi). u = 5, a = -8, s = 1 ii). u = 3, a = -3.4, s = 2 iii). u = 3.5, a = -4.6, s = 5 1

2 4 5 4iv). u = 1 3, a = -7.4, s = 3 v). u = 3.6, a = -2.5, s = 7 vi). u = 2 1, a = -3.9, s = 4 2

4 5 8 4 3

6). Find the value of b - 4ac2 , if

i). a = 3, b = 2 1 , c = -2 ii). a = 3, b = -8, c = -4 iii). a = 1, b = 9, c = -3 1 4 2 5 5 4

iv). a = -1 5, b = 6, c = 4.2 v). a = -5.6, b = 6.1, c = -7 vi). a = 1 1, b = -1.9, c = -3 2 6 9 3 3

7). Find the value of ab2 + b , if a

i). a = 4, b = -3 ii). a = 1, b = -8 iii). a = -1, b = 4 iv). a = -7, b = 2 5 2 5 3

v). a = 1 3, b = -2 vi). a = -5.6, b = -3.1 vii). a = 1 1, b = -1 viii). a = -1 , b = -64 2 3 4


Substitution (Negative Numbers and Fractions).

1). Using the formula v = u + at, find the value of v ifi). u = 2, a = -6, t = 1 ii). u = 10, a = -12, t = 2 iii). u = 20, a = -4, t = 2 1

2 3 2iv). u = -12, a = 9, t = 2 v). u = -4, a = -25, t = 2 vi). u = 14, a = -20, t = 1 1

3 5 4

2). Using the formula y = mx + c, find the value of y ifi). c = -3, x = 9, m = 1 ii). c = -2, x = 8, m = 3 iii). c = -8, x = 4, m = 3 1

2 4 2iv). c = 14, x = -24, m = 3 v). c = -2, x = -36, m = 1 vi). c = -5, x = -4, m = 1 3

4 4 4vii). c = -8, x = 15, m = -2 viii). c = 4, x = -30, m = -2 ix). c = -2, x = -6, m = -1 2

3 5 3

3). Using the formula a = v2 - v

1 , find the value of a (acceleration) if

ti). v

1 = 6, v

2 = 9, t = 1 ii). v

1 = 8, v

2 = 14, t = 1 iii). v

1 = 10, v

2 = 18, t = 2

2 5 3iv). v

1 = 7, v

2 = 2, t = 1 v). v

1 = 2.1, v

2 = 0.7, t = 1 vi). v

1 = 3.4, v

2 = 0.8, t = 2

4 3 3

4). Using the formula s = ut + 1 at2 , find the value of s if 2

i). u = 1, a = 4, t = 6 ii). u = 12, a = 3, t = 10 iii). u = 40, a = -36, t = 1 2 5 2

iv). u = 4, a = -2, t = 5 v). u = 3, a = -3, t = 8 vi). u = 12, a = -18, t = 1 5 4 3

5). Using the formula v2 = u2 - 2as, find the value of v ifi). u = 1, a = -8, s = 1 ii). u = 3, a = -20, s = 2 iii). u = 6, a = -8, s = 1 3

2 5 4iv). u = 7, a = -20, s = 4 v). u = 5, a = -20, s = 3 vi). u =10, a = -8, s = 2 3

5 5 4

6). Find the value of b - 4ac2 , if

i). a = 1, b = -5, c = 9 ii). a = -3, b = 4, c = 3 iii). a = -5, b = 7, c = 3 4 4 4

iv). a = 1 1, b = 10, c = 6 v). a = -10, b = 9, c = -4 vi). a = 1 1, b = 11, c = -92 5 3

7). Find the value of ab2 + b , if a

i). a = 1, b = -4 ii). a = 1, b = -6 iii). a = -1, b = 10 iv). a = -1, b = -9 2 3 2 3


Linear Functions Revision.

The general formula for a linear function is y = mx + cwhere m is the gradient and c is the y intercept.

A. For each of the following functions find

a). the gradient of the line, b). the y intercept.

Do not plot any of the functions.

1). y = 5x + 2 2). y = 1/5 x + 4 3). y = 2x - 5 4). y = 1/

4 x - 3

5). f(x) � -2x + 4 6). y = -3x 7). f:x � 4x - 1 8). f(x) � -1/2 x + 4

9). y = -3x + 2 10). f:x � 3/4 x - 2 11). f:x � -2/

5 x + 7 12). y = -3x + 2.4

13). y = 6 - 3x 14). y = -5 + 2/3 x 15). y = 4 - x 16). y = 1.5 - 2x

17). y = 2.7 + 3x 18). f(x) � -3 - 2x 19). y = 2/3 + 6x 20). f:x � -5 + x

21). f:x � 4/5 x - 1/

322). f(x) � -11/

3 + x 23). y = x - 21/

424). y = -2/

7 x

25). y = 31/2 - 23/

4x 26). f:x � 6.5x 27). f(x) � 54/

5 x - 2 28). y = 4/

5 - x

B. Rearrange the following linear functions with y the subject of the equation.

State i). the gradient of the line, ii). the y intercept.

Do not plot any of the functions.

1). 5x + 3 - y = 0 2). 0 = 6 - y - 2x 3). x + 3 - y = 0 4). 0 = 0.5x - y - 45). y - 4/

5x - 1 = 0 6). 5 - 2x + y = 0 7). 3/

4x - 5 + y = 0 8). 2 + y - 7x = 0

9). 0 = y + 4x - 1/3

10). 0 = 6 + y - 2/3x 11). 0 = 6x + y - 1 12). 0 = 1 - x + y

13). y = 2x 14). y = -4x 15). y = -x 16). 2x = y3 2 4 2 3 9

17). 2x = 4y 18). 3y = -6x 19). 6y = x 20). 5y = 1.25x5 4 7 3.5 6

21). y - x = 0 22). 0 = y - 4x 23). 2.5x - y = 0 24). y + 0.5x = 02 3 2 3

25). 4.8x = 2y 26). 5y = -1.25x 27). 2y = x 28). -1.4x = 7y 3 8 9 6 5

29). 2x - 3 - 3y = 0 30). 0 = 2 - 4y - 3x 31). 0 = 8 - 2x - 5y 32). x - 5y + 6 = 033). 2x + 3y = 9 34). 4y - 8x = 5 35). 3 = 6y - 10x 36). -9 = 8y - 4x37). 3y - 5 - 4x = 0 38). 7 + 2y - 3x = 0 39). 1 - 3x + 5y = 0 40). 0 = 18 + 6x + 8y

C. Plot the following linear functions for values -5 ≤ x ≤ 5.

1). y = x + 2 2). y = 1/2 x - 3 3). y = 2x - 1 4). y = 1/

3 x +4

5). f(x) � 4x + 2 6). y = -x 7). f:x � 3/4 x + 1 8). f(x) � -2/

3 x - 2

9). 2x - 3 - y = 0 10). 0 = 3 - y - 0.75x 11). y + 4x - 1 = 0 12). 2 - 3x + y = 013). y = -0.6x 14). y = 2x 15). 3y = x 16). 3y = -x

2 0.5 4 10 517). 2x + 6 - 3y = 0 18). 6x + 2y = -3 19). 2 - 4x + 5y = 0 20). 0 = -5 + 9x + 4y


D. The gradient of a straight line is

Change in y = y2 - y

1

Change in x x2 - x

1

E.g.1 Find the gradient between (2,4) and (5, 19).

The gradient of a straight line is Change in y = y2 - y

1 = 19 - 4 = 15 = 5

Change in x x2 - x

15 - 2 3

E.g.2 Find the gradient between (-3, 2) and (5, 4).


1 = 4 - 2 = 2 = 1

Change in x x2 - x

1 5 -- 3 +8 4

Find the gradient of the line between these sets of coordinates. (Draw a diagram if it helps you).

1). (1,2), (5, 10) 2). (4,3), (6,9) 3). (4,7), (16,13) 4). (2,1), (14,4)5). (-2,4), (4,8) 6). (-2,7), (0,15) 7). (-4,-4), (-1,11) 8). (-7,1), (-1,9)9). (4,-4), (9, 1) 10). (7,2), (13,10) 11). (-3,-8), (2,7) 12). (3,-3), (11,17)13). (-6,-5), (6,-1) 14). (-2,-13), (0,-1) 15). (-10,-11), (-2,7) 16). (-8,-24), (-2,-2)

E.g.3 Find the gradient between (-1,8) and (1,2).


1 = 2 - 8 = -6 = -3

Change in x x2 - x

1 1 -- 1 +2

Find the gradient of the line between these sets of coordinates.

17). (5,-1), (3,5) 18). (-5,14), (1,2) 19). (-9,-7), (-3,2) 20). (-9,2), (-3,0)21). (-3,-18), (2,2) 22). (-7,-8), (-8,-2) 23). (-24,4), (-8,2) 24). (-6,5), (-1,-20)25). (11,-1), (2,20) 26). (-3,23), (1,5) 27). (-8,-18), (-3,12) 28). (-12,-19), (-4,11)29). (0,2), (-0.5,-1) 30). (6,-14), (9,-2) 31). (-1,-1), (-16,-10) 32). (-5,3), (-15,-11)

E. The mid-point of a line segment joining A (x1,y

1) to B (x

2,y

2) is

(xm,y

m) = x

1 + x

2 , y

1 + y

2

2 2

Find the coordinate of the mid-point joining the following points

1). (2,3) and (8, 11) 2). (4,14) and (8, 4) 3). (3, 0) and (7, 16)4). (4, 7) and (10,13) 5). (0,12) and (8, 4) 6). (1, 0) and (9,16)7). (2, 17) and (16, 5) 8). (7, 2) and (1, 18) 9). (5, 3) and (19, 19)10). (1, 2) and (12, 10) 11). (18, 5) and (0, 12) 12). (7, 17) and (11, 6)13). (0, 17) and (15, 5) 14). (7, 1) and (29, 14) 15). (6, 23) and (24, 6)16). (3, 5) and (14, 22) 17). (4, 13) and (21, 2) 18). (27, 8) and (8, 21)19). (-5, 2) and (1, 14) 20). (7, -8) and (2, 4) 21). (5, 4) and (-5, 15)22). (6, 11) and (15, -9) 23). (-9, 24) and (5, -14) 24). (15, -8) and (-23, 5)25). (7, -7) and (-18, 3) 26). (-12, 5) and (3, -8) 27). (-7, -15) and (13, 8)28). (-2, 4) and (-16, -11) 29). (-26, -15) and (8, -21) 30). (-37, 15) and (-9, -36)


Finding the Equation of a Straight Line.

Draw appropriate axes for each question, using the scales given.Plot the two coordinates given. Join them with a straight line.

Find the equation of the line.

Ques. 1-4. On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit.1). a). (1,6) and (-3,-2) b). (6,10) and (-4,-10) c). (7,10) and (-3,-10).

2). a). (2,7) and (-1,-2) b). (4,9) and (-2,-9) c). (2,6) and (-1,-3).

3). a). (3,-3) and (6,0) b). (5,7) and (-6,-4) c). (7,6) and (-5,-6).

4). a). (8,8) and (-8,0) b). (8,4) and (-8,-4) c). (8,-2) and (-8,-10).

Ques.5. On the x-axis 1 cm = 1 unit, on the y-axis 2 cm = 1 unit.5). a). (8,-1) and (-4,-4) b). (4,2) and (-4,0) c). (6,2) and (-2,0).

Ques.6. On the x-axis 2 cm = 1 unit, on the y-axis 1 cm = 1 unit.6). a). (1,8) and (-3,-8) b). (2,8) and (-2,-8) c). (3,4) and (0,-8).

Ques. 7-9. On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit.7). a). (-3,-9) and (6,9) b). (8,3) and (-6,-4) c). (4,10) and (-3,-11).

8). a). (6,4) and (-3,1) b). (3,11) and (-1,-9) c). (7,6) and (-5,0).

9). a). (4,-7) and (-4,9) b). (0,-3) and (4,-11) c). (-4,10) and (3,-4).

Ques. 10-11. On the x-axis 1 cm = 1 unit, on the y-axis 2 cm = 1 unit.10). a). (6,-1) and (-6,3) b). (6,2) and (-3,5) c). (0,-5) and (-6,-3).

11). a). (8,2) and (2,5) b). (2,-5) and (-4,-2) c). (-1,5) and (7,1).

Ques. 12. On the x-axis 2 cm = 1 unit, on the y-axis 1 cm = 1 unit.12). a). (1,-3) and (-3,9) b). (-3,4) and (2,-11) c). (3,-7) and (-3,11).

Ques. 13-18. On the x-axis 1 cm = 1 unit, on the y-axis 1 cm = 1 unit.13). a). (-8,7) and (8,3) b). (-6,0) and (2,-2) c). (4,2) and (-8,5).

14). a). (1,1) and (-2,10) b). (-6,1) and (3,-2) c). (7,-6) and (-1,10).

15). a). (-6,8) and (4,3) b). (-2,9) and (1,-9) c). (-2,5) and (-6,6).

16). a). (5,7) and (-1,-11) b). (7,-8) and (-2,10) c). (3,9) and (-2,-11).

17). a). (-6,9) and (3,6) b). (8,-1) and (-4,-7) c). (7,10) and (-5,4).

18). a). (5,-9) and (-2,12) b). (3,11) and (0,-7) c). (-8,6) and (8,4).

Ques. 19-20. On the x-axis 2 cm = 1 unit, on the y-axis 1 cm = 1 unit.19). a). (-2,8) and (1,-7) b). (-1,11) and (3,9) c). (3,9) and (-3,8).

20). a). (2,-10) and (-2,10) b). (3,-4) and (-3,-3) c). (3,7) and (-1,-13).


Finding Equations of Straight Line Graphs 1.

Copy the diagrams onto squared paper, then find the equation of each line.1). 2). 3).

4). 5). 6).

7). 8). 9).

10). 11). 12).

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

a.b.c.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

a.

c.

b.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

a.

b.

c.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

a.b.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

a.

b.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.


Finding Equations of Straight Line Graphs 2.

These are now a little harder, be careful.

1). 2). 3).

4). 5). 6).

7). 8). 9).

10). 11). 12).

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

64

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b.

a.

6

6

4

4

2

2

-2

-2

-4

-4

-6

-6

c.

b. a.


Copy the diagrams onto squared paper.Find the equation of each line.

Be careful of the scales.13). 14). 15).

16). 17). 18).

19). 20). 21).

Very difficult.22). 23). 24).

6

3

4

2

2

1

-1

-2

-2

-4

-3

-6

c.

b.a.

3

6

2

4

1

2

-2

-1

-4

-2

-6

-3

c.

b.

a.

6

12

4

8

2

4

-4

-2

-8

-4

-12

-6

c. b.

a.

12

3

8

2

4

1

-1

-4

-2

-8

-3

-12

c.

b.

a.

6

3

4

2

2

1

-1

-2

-2

-4

-3

-6

c.b.

a.

12

6

8

4

4

-2

-4

-4

-8

-6

-12

c.

b.

a.

2

6

12

4

8

2

4

-4

-2

-8

-4

-12

-6

c.

b.

a.

3

12

2

8

1

4

-4

-1

-8

-2

-12

-3

c.

b.

a.

3

6

2

4

1

2

-2

-1

-4

-2

-6

-3c.

b.a.

12

3

8

2

4

1

-1

-4

-2

-8

-3

-12

c.b.

a.

12

6

8

4

4

2

-2

-4

-4

-8

-6

-12

c.

b.

a.

3

12

2

8

1

-4

-1

-8

-2

-12

-3

c.

b.

a.4


y = x y = x + 2 y = x - 4

y = -x y = -x + 2 y = -x - 4

y = 1 x y = 1 x + 2 y = 1x - 4 2 2 2


y = - 3x y = -3x + 2 y = -3x - 4

y = 3x y = 3x + 2 y = 3x - 4

y = -1x y = -1x + 2 y = -1x - 4 2 2 2


2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X


2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X

2

4

6

8

10

12

2 4 6 8 -8 -6 -4 -2 -2-4

-6-8

-10

-12

Y

X


Scatter Graphs 1.

1). 10 pupils sat both Maths and Physics exams, here are their scores:

A B C D E F G H I JMaths 56 24 67 70 71 42 48 32 52 80Physics 65 38 71 72 73 51 56 42 57 82

a). Plot them as a scatter graph and comment on the type of correlation shown.b). Draw in the line of best fit.c). Use your graph to answer these questions, showing any construction lines used.

i). One pupil scored 65 % in the Maths exam, but was absent for the Physics exam.What would be an appropriate estimate for that pupil's Physics examination ?

ii). A pupil scored 39 % in the Physics exam, give an estimate for the Maths exam score ?

2). 12 pupils sat both Science and History exams, here are their scores:

A B C D E F G H I J K LScience 15 63 18 34 44 50 25 54 85 29 39 74History 72 38 72 58 52 50 66 44 19 63 54 28


i). One pupil scored 60 % in the Science exam, but was absent for the History exam.What would be an appropriate estimate for that pupil ?

ii). A pupil scored 60 % in the History exam, what do you think the pupil got in the Science exam ?

3). Here are the scores of 10 people who took an Intelligence test. The I.Q. was measuredas a percentage of the test. Their ages were also recorded.

A B C D E F G H I JAge of person 22 41 83 30 55 62 72 39 26 65I.Q. % 73 36 66 47 96 64 41 64 91 88

a). Plot them as a scatter graph and comment on the type of correlation shown.b). If a 50 year old took the test, could you predict from the graph the score the person would get ?

4). A Biologist took measurements from a selection of 12 Beech leaves.

A B C D E F G H I J K LWidth (mm) 28 25 16 32 40 25 22 11 36 15 19 4Length (mm) 49 46 27 55 68 18 37 21 61 28 34 8


i). If the width of a leaf is 35 mm, give a good estimate of the length of the leaf.ii). If the length of a leaf is 30 mm, give a good estimate of the width of the leaf.

d). When conducting the measurements the Biologist made a mistake measuring one of the leaves.Which leaf was it ?


5). Here are the scores given to 12 entrants in the "champion leek" competition.

A B C D E F G H I J K LJudge 1 70 25 50 15 72 25 5 40 80 60 35 54Judge 2 85 30 55 10 88 26 1 71 95 69 40 65

a). Plot them as a scatter graph and comment on the type of correlation shown.b). Draw in the line of best fit.c). i). A late entrant scored 65 marks from Judge 1, but Judge 2 had gone home.

What would be an appropriate mark for that entrant ?ii). Judge 2 gave last years winner 82 marks. What would judge 1 have given last years winner ?

d). One of the leeks got misjudged due to a technical error. Which leek do you think it was ?

6). A basketball coach wonders if there is a link between the number of points a player scores and thenumber of fouls a player gives away in a match. Here are 10 players scores for one match.

A B C D E F G H I JFouls against 3 5 15 9 42 22 45 6 36 27Points scored 1 0 22 9 72 30 85 3 57 54

a). Plot them as a scatter graph and comment on the type of correlation shown.b). Draw in the line of best fit.c). i). One player scored 50 points, how many fouls do you think he picked up ?

ii). One player gave away 12 fouls, how many points do you think he scored ?

7). A hotdog vendor recorded the number of hotdogs sold in a day (to the nearest 10) and themaximum temperature that day (˚C). Here is a fair selection of results.

Hotdogs sold 610 250 90 780 400 130 420 850 680 510 270 690Temperature ˚C 14 32 39 9 27 34 22 4 35 20 27 16

a). Plot them as a scatter graph and comment on the type of correlation shown.b). Draw in the line of best fit.c). i). How many hotdogs would he expect to sell when the maximum temperature is 30˚C that day ?

ii). If he sold 700 hotdogs in a day what do you think the maximum temperature was ?d). One of the days was National Hotdog Day, when the public was encouraged to buy more hotdogs.

Which set of results do you think show this day ?

8). 12 pupils sat Physics and Chemistry exams. Here are their percentage scores:

Physics 55 44 57 73 59 49 50 35 47 51 61 39Chemistry 50 37 55 70 58 60 44 29 44 49 60 32

a). Plot them as a scatter graph and comment on the type of correlation shown.b). Draw in the line of best fit.c). i). A pupil scored 40 % in the Chemistry exam, but was absent for the Physics exam.

What would be an appropriate estimate for that pupil ?ii). A pupil scored 65 % in Physics, what do you think they got in the Chemistry exam ?

d). One pupil felt very ill for the Physics exams, but fine for Chemistry. Which pupil was it ?


Scatter Graphs 2.

1). A speedometer measuring Km/h is found to be faulty. It is then tested to find out thereadings that it gives against the true speed of the car. Here are the test results.

True speed, x 80 8 23 40 32 54 73 67 17 56Speedometer Reading, y 71 19 28 44 38 52 68 60 26 54

a). Plot them as a scatter graph.b). Calculate the mean of :-

i). the true speed of the car, x ,ii). the speedometer readings, y.

c). Plot (x, y) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). i). If the true speed of the car is 30 Km/h, what would be a good estimate for the reading

that the speedometer would show ?ii). If the speedometer shows 65 Km/h, what is a good estimate of the true speed ?

2). 10 pupils sat Paper 1 and Paper 2 of a Physics exam. Here are their results :

Physics P1, x 60 50 30 46 25 36 31 19 43 20Physics P2, y 55 49 34 44 30 40 36 26 41 25


i). the Paper 1 results, x,ii). the Paper 2 results, y.

c). Plot (x, y) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). i). One pupil scored 55 % in Paper 1, but was absent for Paper 2.

What would be an appropriate estimate for Paper 2 for that pupil ?ii). A pupil scored 20 % in Paper 2, what do you think the pupil got in Paper 1?

3). A business woman records the length of time it takes to get to work in a morning and themaximum speed on her journey.

Journey time (minutes) 35 10 70 48 23 22 51 44 29 56Maximum speed (Km/h) 50 67 20 36 55 60 32 42 50 34


i). the journey time, x,ii). the maximum speed, y.

c). Plot (x, y) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). i). One day the journey time was 60 minutes. What would be an appropriate estimate for

the maximum speed she reached on that journey ?ii). Another day the maximum speed she reaches is 38 Km/h.

What would be a good estimate of that journey's time ?


4). 10 pupils sat a Physics and Chemistry exam. Here are their percentages:

Physics, P 10 70 52 32 79 22 45 63 12 55Chemistry, C 19 82 60 43 88 31 54 73 23 67


i). the Physics exam, P , ii). the Chemistry exam, C.c). Plot (C, P) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). i). One pupil scored 40 % in the Physics exam, but was absent for the Chemistry

exam. What would be an appropriate estimate for that pupil ?ii). A pupil scored 80 % in the Chemistry exam, what do you think they got in the

Physics exam ?f). The equation of the line of best fit can be written as C = aP + b,

where a and b are numbers. Find the values of a and b.g). Use your formula to find the value of C when

i). P = 24%, ii). P = 72%, iii). P = 37%.

5). 10 pupils sat a Maths and Physics exam. Here are their percentages:

Maths, M 15 71 60 35 25 47 85 19 79 34Physics, P 40 72 64 55 46 60 75 46 72 50


i). the Maths exam, M , ii). the Physics exam, P.c). Plot (M, P) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). i). One pupil scored 65 % in the Maths exam, but was absent for the Physics

exam. What would be an appropriate estimate for that pupil ?ii). A pupil scored 41 % in the Physics exam, what do you think they got in the

Maths exam ?f). The equation of the line of best fit can be written as P = aM + b,

where a and b are numbers. Find the values of a and b.g). Use your formula to find the value of P when

i). M = 90%, ii). M = 96%, iii). 100%.

6). Here are the equivalent British and Continental shoe sizes.

British, B 1 2 3 4 5 6 7Continental, C 33 34 36 37 38 39 41

a). Plot them as a scatter graph.b). Calculate the mean of the :-

i). British Shoe sizes, B , ii). Continental shoe sizes, C.c). Plot (B, C) on the scatter graph and indicate this point.d). Draw in the line of best fit and comment on the type of correlation shown.e). The equation of the line of best fit can be written as C = aB + b,

where a and b are numbers. Find the values of a and b.f). Use your formula to find the value of C when

i). B = 0, ii). B = 10, iii). B = 15.


Direct Proportion.

There is a direct proportion between two variables when one variable isa multiple of the other, i.e. their ratio is a constant.

There is a direct proportion between the perimeter of a square and its lengthP = 4l, there is a multiplying factor of 4.

There is a direct proportion between the circumference of a circle and its radius.C = 2πr, there is a multiplying factor of 2π.

There is a direct proportion between the miles and kilometres.1 mile = 1.6 kilometres, there is a multiplying factor of 1.6.

The symbol for proportion is ∝. So the statement:-

'The circumference of a circle is directly proportional to its radius' could be written as

C ∝ r which implies thatC = kr, where k is the constant of proportionality.

The first step in any question of this type is to find a value for k. In this case it is 2π.

1). y is directly proportional to x. y = 20 and x = 4.a). Write an equation in y and x.b). Find k, the constant of proportionality.c). Hence, write a new equation in y and x.

2). P is directly proportional to V. P = 15 and V = 5.a). Write an equation in P and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in P and V.

3). S is directly proportional to T. S = 21 and T = 6.a). Write an equation in S and T.b). Find k, the constant of proportionality.c). Hence, write a new equation in S and T.

4). p is directly proportional to q. p = 47 and q = 10.a). Write an equation in p and q.b). Find k, the constant of proportionality.c). Hence, write a new equation in p and q.

5). U is directly proportional to V. U = 8 and V = 16.a). Write an equation in U and V.b). Find k, the constant of proportionality.c). Hence, write a new equation in U and V.

6). a is directly proportional to b. a = 1.8 and b = 9.a). Write an equation in a and b.b). Find k, the constant of proportionality.c). Hence, write a new equation in a and b.

7). V is directly proportional to W. V = 12 and W = 18.a). Write an equation in V and W.b). Find k, the constant of proportionality.c). Hence, write a new equation in V and W.


E.g. The cost of making a box is directly proportional to the amount of time spent making it.It took 5 hours to make a box costing £20. Find :-a). the cost of a box that took 7 hours to make,b). the time it took to make a box costing £18.

C ∝ t, so C = kt , where k is the constant of proportionality.

When C = £20, t = 5 hours.20 = k x 5, so k = 4. C = 4t.

a). C = 4 x 7 C = £28 b). 18 = 4 x t t = 4.5 hours.

8). T is directly proportional to U. If T = 24 when U = 8, find :-a). T when U = 7, b). U when T = 39.

9). p is directly proportional to q. If p = 43.2 when q = 7.2, find :-a). p when q = 9, b). q when p = 69.

10). A is directly proportional to B. If A = 38.4 when B = 16, find :-a). A when B = 22, b). B when A = 81.6.

11). f is directly proportional to d. If f = 69.6 when d = 12, find :-a). f when d = 14, b). d when f = 145.

12). T is directly proportional to U. If T = 13 when U = 26, find :-a). T when U = 47, b). U when T = 35.

13). r is directly proportional to s. If r = 9 when s = 12, find :-a). r when s = 60, b). r when s = 99.

14). f is directly proportional to g. If f = 125.8 when g = 34, find :-a). f when g = 18, b). g when f = 127.65.

15). C is directly proportional to D. If C = 138.6 when D = 16.5, find :-a). C when D = 9.5, b). D when C = 163.8.

16). If the cost of 12 apples is £1.68.a). Write a formula linking the cost and the number of apples.b). Find the cost of 23 apples.c). How many apples would you get for £5.18 ?

17). The distance covered by a car is directly proportional to the time taken.The car covers 91 km in 3.5 hours.a). Find how far it covers in 2.5 hours.b). If it travels 110.5 km, how long has it been travelling ?

18). The cost of using the internet with an I.S.P.'s is directly proportional to the time spent onthe internet. It costs 12.5p for 50 minutes.a). How much will it cost to spend 1 hour 10 minutes on the internet ?b). If the cost is £2.10, how long has been spent on the internet :-

i). in minutes, ii). in hours ?19). A sign writer charges customers a cost directly proportional to the area of the sign.

A sign 2.4 m2 costs £86.40.a). How much will it cost for a 6.3 m2 ?b). Another sign cost £32.40 , what is the area :-

i). in m2, ii). in mm2 ?20). The number of children who can be safely accommodated in a classroom

is directly proportional to the floor area of the classroom.A classroom with floor area 27.6m2 is safe for 12 children.a). What is the floor area of a classroom need to safely hold 32 children ?b). A classroom has a floor area of 50 m2, what is the maximum number of children it

can safely hold ?


Quadratic Sequences.

Linear Sequences (Revision).

1). For each of the following linear sequencesi). write the next two terms, ii). find the nth term.

a). 2 4 6 8 10 b). 6 12 18 24 30 c). 4 7 10 13 16d). 2 6 10 14 18 e). 1 4 7 10 13 f). 2 7 12 17 22g). 2 8 14 20 26 h). -5 -2 1 4 7 i). -17 -13 -9 -5j). -11 -5 1 7 k). 15 13 11 9 7 l). 35 32 29 26 23m). 4 3 2 1 0 n). 27 23 19 15 o). 3 1 -1 -3p). -5 -7 -9 -11 -13 q). 51/

2 6 61/

2 7 71/

2r). 31/

4 31/

2 3 3/

4 4 41/

4

s). 3/5

4/5 1 1 1/

5 1 2/

5t). 22/

3 21/

3 2 1 2/

3

u). What do you notice about the differences between the numbers for each linear sequence ?

2). For each of the following linear sequencesi). find the nth term, ii). find the 30th term.

a). 3 5 7 9 11 b). 1 5 9 13 17 c). 2 5 8 11 14d). -1 3 7 11 15 e). -2 4 10 16 22 f). 4 9 14 19 24g). -1 5 11 17 23 h). 1 9 17 25 33

3). These are harder. For each of these following sequencesi). write the next two terms, ii). find the nth term.

a). 1 1 1 1 1 b). 1 1 1 1 1 c). 1 2 3 4 52 6 10 14 18 7 10 13 16 19 3 5 7 9 11

d). 1 2 3 4 5 e). -1 -2 -3 -4 -5 f). 2 4 6 8 106 7 8 9 10 1 4 7 10 13 1 5 9 13 17

g). 3 6 9 12 h). 1 3 5 7 93 7 11 15 5 8 11 14 17

4). Write the first five terms of the linear sequence which has its nth term asa). n + 4 b). 2n + 3 c). 3n - 3 d). 4n - 1 e). 2n - 3f). 3n - 7 g). 5n - 11 h). n__ i). n + 2 j). 3n - 1

n + 1 2n + 1 5n + 2Quadratic Sequences.

1). For each of the following quadratic sequencesi). write the next two terms, ii). explain how you found them.

a). 7 8 10 13 17 b). 2 3 5 8 12 c). 1 3 7 13 21d). 5 8 14 23 35 e). 4 10 22 40 64 f). 2 7 17 32 52g). 5 9 17 29 45 h). 3 10 24 45 73 i). 3 13 33 63 103j). 7 15 31 55 87 k). 63 61 57 51 43 l). 82 78 70 58 42m). 92 89 83 74 62 n). 200 193 179 158 130

o). What do you notice abouti). the differences between the numbers for each quadratic sequence ?ii). the second difference row for each quadratic sequence ?

2). These are square numbers. 1 4 9 16 25 36 49 .........What is the nth term of square numbers ?


3). All the following sequences are based on the square numbers. To solve the following questionscompare the square numbers with the sequences you are given.For each of the following sequences

i). write the next two terms, ii). find the nth term.a). 2 5 10 17 26 b). 6 9 14 21 30 c). 0 3 8 15 24d). -3 0 5 12 21 e). 2 8 18 32 50 f). 4 16 36 64g). 3 12 27 48 75 h). 1/

2 2 41/

2 8 121/

2i). 1/

4 1 21/

4 4 61/

4

j). 3 5 7 9 11 k). 1 4 9 16 251 4 9 16 25 5 8 11 14 17

l). 1 2 3 4 5 m). 4 7 10 13 162 5 10 17 26 2 8 18 32 50

Difficult n). 2 6 12 20 30 o). 0 2 6 12 20

4). Write the first five terms of the quadratic sequence which has its nth term asa). n2 + 2 b). n2 - 3 c). 5n2 d). n2 + 2n e). 2n2 + n + 1

5). Here are 6 terms in a sequence of numbers 0 2 6 12 20 30.a). i). Write the next number, ii). explain how you found it.b). The pattern below gives the same sequence.

Term 1 Term 2 Term 3 Term 4 Term 5 Term 6 1 x 0 2 x 1 3 x 2 4 x 3 5 x 4 6 x 5i). Find the 20th term. ii). Find the nth term.





8). Here is a sequence of numbers 4 10 18 28 40a). Find the next term. b). The nth term is n2 + bn, where b is a number. Find the value of b.

9). Here is a sequence of numbers 5 12 21 32 45a). Find the next term. b). The nth term is n2 + bn, where b is a number. Find the value of b.

10). Here is a sequence of numbers 11/2 5 101/

2 18 271/

2

a). Find the next term. b). The nth term is n2 + bn, where b is a number. Find the value of b.

11). Here is a sequence of numbers 4 14 30 52 80a). Find the next term. b). The nth term is an2 + n, where a is a number. Find the value of a.

12). These sequences are not quadratic.Find the nth term for these sequencesa). 1 8 27 64 b). 2 9 28 65 c). 2 16 54 128


Finding Quadratic Functions (The Difference Method).

1). Copy and complete the table below for y = 1 x2 .

x 1 2 3 4 5y 1 4 9

1st Difference row 3 5 2nd Difference row 2

Look at the 2nd difference row. What do you notice ?

2). Copy and complete the table for y = 2x2.

x 1 2 3 4 5y 2 8

Fill in the difference rows. Look at the 2nd difference row. What do you notice ?

3). Repeat question 2 fora). y = 2x2 + 3 b). y = 3x2 + x, c). y = 4x2 - 1, d). y = 5x2 + 2.

4). In each question above, compare the coefficient of x2 (the number in front of x2) and thenumber in the second difference row. What do you notice ?

How to find a Quadratic Equation using the Difference Method.

The general equation for any quadratic is

y = ax2 + bx + c , where a, b and c are numbers.

E.g. Find the equation that links x and y in the table.

x 1 2 3 4 5 6y 4 13 26 43 64 89

First look at the difference rows.

x 1 2 3 4 5 6y 4 13 26 43 64 89

1st Diff. row 9 13 17 21 252nd Diff. row 4 4 4 4 This tells us we have a quadratic equation.

It also tells us the coefficient of x2 is 2 (1/2 of 4).

Hence we now know y = 2x2 + bx + c.

We now take the 2x2 out of the table. This will leave the rest of the equation.

x 1 2 3 4 5 6y 4 13 26 43 64 89 _2x2 2 8 18 32 50 72

bx + c 2 5 8 11 14 17 This is the missing part of the equation.


Putting x and bx + c x 1 2 3 4 5 6in to a table. bx + c 2 5 8 11 14 17

Difference 3 3 3 3 3 Hence bx+ c = 3x - 1.

Therefore our quadratic equation is y = 2x2 + 3x - 1 Finding quadratic equations this way will only be required in coursework.

Find the formula that link each of these quadratic functions.

1). x 1 2 3 4 5 6 16). x 1 2 3 4 5 6 y 7 16 29 46 67 92 y 4 6 9 13 18 24

2). x 1 2 3 4 5 6 17). x 1 2 3 4 5 6 y 7 21 43 73 111 157 y 6.5 13 23.5 38 56.5 79

3). x 1 2 3 4 5 6 18). x 1 2 3 4 5 6 y 8 18 34 56 84 118 y 3.5 8.5 15.5 24.5 35.5 48.5

4). x 1 2 3 4 5 6 19). x 1 2 3 4 5 6 y 3 7 15 27 43 63 y 3.25 4 5.25 7 9.25 12

5). x 1 2 3 4 5 6 20). x 1 2 3 4 5 6 y 3 12 25 42 63 88 y 1.25 3.25 6.25 10.25 15.25 21.25

6). x 1 2 3 4 5 6 21). x 1 2 3 4 5 6 y 5 12 25 44 69 100 y 2.25 8.75 19.25 33.75 52.25 74.75

7). x 1 2 3 4 5 6 22). x 1 2 3 4 5 6 y -3 4 17 36 61 92 y 0.75 2 3.75 6 8.75 12

8). x 1 2 3 4 5 6 23). x 1 2 3 4 5 6 y 5 11 21 35 53 75 y 0.75 3.25 7.75 14.25 22.75 33.25

9). x 1 2 3 4 5 6 24). x 1 2 3 4 5 6 y 2 12 30 56 90 132 y 0.35 1.2 2.55 4.4 6.75 9.6

10). x 1 2 3 4 5 6 25). x 1 2 3 4 5 6 y 4 15 32 55 84 119 y 2 3.5 6 9.5 14 19.5

11). x 1 2 3 4 5 6 26). x 1 2 3 4 5 6 y 1 10 23 40 61 86 y 4 8 15 25 38 54

12). x 1 2 3 4 5 6 27). x 1 2 3 4 5 6 y -5 -3 3 13 27 45 y -0.5 0.5 2.5 5.5 9.5 14.5

13). x 1 2 3 4 5 6 28). x 1 2 3 4 5 6 y 4 12 22 34 48 64 y 2.5 8.5 17.5 29.5 44.5 62.5

14). x 1 2 3 4 5 6 29). x 1 2 3 4 5 6 y 1.5 6 11.5 18 25.5 34 y 3.95 15.65 35.35 63.05 98.75 142.45

15). x 2 3 4 5 6 7 30). x 1 2 3 4 5 6 y -4 -3.5 -2 0.5 4 8.5 y 3.05 11.35 25.65 45.95 72.25 104.55


Practical Quadratic Number Patterns.

1). a). Copy the diagrams.b). Copy and complete the table below

Width, x 1 2 3 4 5Number of squares, s 1 4

c). Find the equation that links the width with the number of squares used.d). Use your formula to find the number of squares used if the width is

i). 7 ii). 15 iii). 24 iv). 37.e). By continuing the table, or otherwise, find the width if the number of squares used is

i). 144 ii). 196 iii). 289 iv). 400.


Number of triangles, t 1 2 3 4 5Number of matches, m 3 12

c). Find the equation that links the number of triangles with the number of matches used.d). Use your formula to find the number of matches used if the number of triangles is

i). 6 ii). 10 iii). 13 iv). 20.e). By continuing the table, or otherwise, find the number of triangles if the number of

matches used isi). 147 ii). 243 iii). 588 iv). 867.


Base width, b 1 2 3 4 5Number of matches, m 3 8

c). Find the equation that links the base width with the number of matches used.d). Use your formula to find the number of matches used if the base width is

i). 6 ii). 12 iii). 15 iv). 24.e). By continuing the table, or otherwise, find the base width if the

number of matches used isi). 80 ii). 120 iii). 195 iv). 288.


Width, x 1 2 3 4 5Number of matches, m 4 10

c). Find the equation that links the width with the number of matches.d). Use your formula to find the number of matches used if the width is

i). 7 ii). 13 iii). 20 iv). 50.e). By continuing the table, or otherwise, find the width if the number of matches used is

i). 54 ii). 130 iii). 180 iv). 418.




c). Find the equation that links the width with the number of matches used.d). Use your formula to find the number of matches used if the width is


i). 161 ii). 320 iii). 616 iv). 1121.


Base width, b 1 2 3 4 5Number of squares, s 1 3

c). Find the equation that links the base width with the number of squares.d). Use your formula to find the number of squares used if the base width is

i). 7 ii). 14 iii). 18 iv). 35.e). By continuing the table, or otherwise, find the base width if the number of

squares used isi). 21 ii). 55 iii). 91 iv). 210.



c). Find the equation that links the width with the number of matches used.d). Use your formula to find the number of matches used if the width is


i). 127 ii). 241 iii). 337 iv). 1057.


Width, x 1 2 3 4 5Number of matches, s 4 13

c). Find the equation that links the width with the number of matches.d). Use your formula to find the number of matches used if the width is

i). 8 ii). 15 iii). 30 iv). 64.e). By continuing the table, or otherwise, find the width if the number of squares used is

i). 89 ii). 274 iii). 559 iv). 1126.


AwayA B C

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Home

Investigations. (Quadratic Patterns).

1). a). Ten teams agree to play each other in a football tournament.Each team has to play each other home and away.Last year only three teams took part.Here is how the league organised it.

How many matches need to be played this year with 10 teams in the tournament ?Next year 15 teams have agreed to take part.How many matches need to be played with 15 teams in the tournament ?Find an expression for the number of matches to be played with n teams in thetournament ?

b). The pools panel look at all the matches that have been a draw at full time.They have to decide what the half time score could have been.If the match was a 2 - 2 draw then there are 9 possibilities that the half time score

could could have been:0 - 0, 1 - 0, 0 - 1, 1 - 1, 2 - 0, 0 - 2, 2 - 1, 1 - 2, 2 -2.Write all the possibilities of the half time scores if the final score wasi). 0 - 0, ii). 1 - 1, iii). 3 - 3, iv). n - n.

c). The pools panel now look at the number of goals scored intotal at the end of each match.Again they are looking at all the possible half time scores.If a match has 2 goals in it there are 6 possible half time scores:0 - 0, 1 - 0, 0 - 1, 1 - 1, 2 - 0, 0 - 2.Write all the possibilities of the half time score if the following numbers of goalswere scored in the match:i). 0, ii). 1, iii). 3, iv). n.

d). At the awards ceremony at the end of the tournament thetrophies are arranged in 3 layers on the presentation table asshown. All are to be given out.How many trophies are given out this year ?Next year it is anticipated there will be 6 layers of trophiesarranged in the same way. How many will be given out nextyear ?If there are n layers of trophies, find an expression for the number of trophiesthere are ?

e). To collect the trophies a podium is built out of blocks.The podium used is 3 layers high and uses 12 blocks,as shown in the diagram.Last year the podium was only 2 layers high. Howmany blocks were needed last year ?Next year the podium is to be 5 layers high. How many blockswill be needed for this podium ?If the podium is n layers high find an expression for the number blocks that there are.


2). When onions have been gathered from the garden they are laidout on the floor to dry.Keith lays out his onions in a square grid as shown.Unfortunately if any onions are rotten the rot will spread.The rot spreads to onions immediately next to them,but not diagonally next to them.It takes one week for rot to spread to the next onion.The diagrams below show the spread of rotten onions over a 2 week period.

Week 0 Week 1 Week 2

Rotten onionsshown by blackcircles.

Carry on the pattern.Find a formula that shows the number of rotten onions after n weeks.Vary the starting number of rotten onions in a row (as below).

For each one find a formula in n for thenumber of rotten onions after n weeks.

Find a link between eachof these formulae.

Experiment with different starting positions for the rotten onions.Use squares, triangles, L shapes etc..

3). When a triangle has no additional lines drawn from the apexto the base, there is only 1 triangle present.When one line is drawn from the apex to the base there are 3 triangles.

How many triangles are created when 2 lines are drawn ?Investigate the relationship between the number of lines drawn from the apex to base andthe number of triangles created.Find a formula for the number of triangles created if there are n lines drawn.

4). A cube of dimensions 3 x 3 x 3 is droppedinto a can of paint and completely covered.Investigate the number of faces covered inpaint on each unit cube.

Unit cube

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Welcome! [smartfuse.s3.amazonaws.com]smartfuse.s3.amazonaws.com/2edcdb4fae2b1d953afcf227a969048e/... · Welcome! This is the first in a series of teaching aids designed by teachers