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Number & Algebra: Strands 3 & 4#1A Relations Approach to Algebra: Linear Functions#2A Relations Approach to Algebra: Quadratic, Cubic & Exponential Functions#3Applications of Sequences & Series#4Applications of Sequences & SeriesName:School:
LINEAR PATTERNS/ARITHMETIC SEQUENCES AND SERIES1.1THE GYM(a)Given that membership to a gym costs 50 and the cost per visit is 10, completethe total cost c in column in following table, showing how a receptionist in this gymwould calculate the cost to a client using the gym:Number of visits tothe gymTotal cost cin Change0123456(b)Represent the data in the above table on a graph.(c)Which is the independent variable and which is the dependent variable? Why?(d)Describe the pattern in your own words.(e)Write down a formula to represent the total cost of the gym in relationship to the numberof visits. State clearly the meaning of any letters used in the formula.(f)Complete the change column in the table in part (b) of this question.(g)What is the rate of change between any points on your graph?(h)What is another name for rate of change?(i)Is there a pattern modelled in the table and graph of this question and if so whattype? Justify your answer.1Project Maths Development Team 2012
1.2TWO MORE GYMSThe following two diagrams models the costs of two other gyms called Gym B and Gym C.Describe in your own words the total cost of each of these gyms and write down a formula torepresent the total cost of each gym in relationship to the number of visits.Gym BGym CModular Course 32
1.3BOX OF SWEETSJoan got a present of a box of 100 sweets. She decides to eat 5 sweets per day from the box.Represent this information in a table, a graph and a formula. After how many days will she haveexactly 40 sweets left?Day NumberSweets in BoxChange01234561.4SPENDING HABITS(a)Given Carol’s and Joan’s spending habits are modelled by the pattern in the diagrambelow describe each of their spending habits?(b)At the beginning of which week will Joan have no money and at the beginning ofwhich week will Carol have no money?3Project Maths Development Team 2012
1.5PATTERNSComplete the next term of the following pattern and complete a table, graph and formula forthe pattern.DiagramNoNo of squares1125Pattern34561.6DVD CLUBJoan joins a DVD club. It costs 12.00 to join the club and any DVD she rents will cost an extra 2. Jonathan joins a different DVD club where there is no initial charge, but it costs 4 to rent aDVD. Represent these 2 situations in a table.Number ofDVDs (x)Cost to Joan (y)Cost to Jonathan (y)0Term1(a)(i)(ii)(iii)List the sequence that represents the cost of the DVDs to Joan.Is this sequence an arithmetic sequence and why?In terms of a the first term, d the common difference and n the number ofthe terms, derive a formula for Tn for this sequence.(b)(i)(ii)(iii)List the sequence that represents the cost of the DVDs to Jonathan.Is this sequence an arithmetic sequence and why?In terms of a the first term, d the common difference and n the number ofthe terms, derive a formula for Tn for this sequence.Modular Course 34
1.7ARITHMETIC SEQUENCEAn arithmetic sequence is such that Tn 4n 3. Is it possible to find three consecutive terms of thissequence such that their sum is equal to 117 and if so find these terms.1.8WORKEmma earns 300 during her first week in the job and each week after that she earns an extra 20 per week. How much does she earn during her tenth week in this job? How much does sheearn in total during the first ten weeks in this job?1.9SUMMING THE NATURAL NUMBERS(i)(ii)Find the sum of the first 100 Natural numbers.Find the sum of all integers, from 5 to 1550 inclusive that are divisible by 5.1.10DESIGNER TILESCaroline bought 400 designer tiles at an end of line sale.She wants to use them as a feature in her new bathroom.If she decides on a pattern of the format shown, how many tiles willbe on the bottom row of the design if she uses all the tiles?1.11BUYING A CARDenise has no savings and wants to purchase a car costing 6000. She starts saving in January2012 and saves 200 that month. Every month after January 2012 she saves 5 more than shesaved the previous month. When will Denise be able to purchase the car of her choice? Ignoreany interest she may receive on her savings.1.12ANALYSING A GRAPHDoes the following graph model a sequence of numbers that form an arithmetic sequence?Explain your reasoning.5Project Maths Development Team 2012
1.13TEACHERS’ SWEETSA teacher is distributing sweets to 120 students. She gives each child a unique number startingat 1 and going to 120. She then distributed the sweets as follows, if the student has an odd ontheir ticket she gives them twice the number of sweets as their ticket number and if the numberon their ticket is even she gives then three times the number of sweets as their ticket number.How many sweets does she distribute?1.14SAVINGSJoe had been saving regularly for some months when he discovered he had lost his savingsrecords. He found two records that showed he saved 260 on the fifth month and had a totalof 3300 on the eleventh month. He knows he increased the amount he saved each month bya constant amount and had some savings before he commence this savings plan. How muchdid he increase his savings by each month and how much had he in his account when hestarted this savings plan?1.15SAMPLE PAPER PHASE 2 LEAVING CERTIFICATE ORDINARY LEVEL PAPER 1 QUESTION 5Síle is investigating the number of square grey tiles needed to make patterns in a sequence. Thefirst three patterns are shown below, and the sequence continues in the same way. In eachpattern, the tiles form a square and its two diagonals. There are no tiles in the white areas in thepatterns – there are only the grey tiles.(a)In the table below, write the number of tiles needed for each of the first five patterns.Pattern12345No. of tiles 2133(b)Find, in terms of n, a formula that gives the number of tiles needed to make the nthpattern.(c)Using your formula, or otherwise, find the number of tiles in the tenth pattern.(d)Síle has 399 tiles. What is the biggest pattern in the sequence that she can make?(e)Find, in terms of n, a formula for the total number of tiles in the first n patterns.(f)Síle starts at the beginning of the sequence and makes as many of the patterns as shecan. She does not break up the earlier patterns to make the new ones. For example,after making the first two patterns, she has used up 54 tiles, (21 33). How many patternscan she make in total with her 399 tiles?Modular Course 36
1.16SAMPLE PAPER PHASE 2 LEAVING CERTIFICATE ORDINARY LEVEL PAPER 1 QUESTION 5John is given two sunflower plants. One plant is 16 cm high and the other is 24 cm high. Johnmeasures the height of each plant at the same time every day for a week. He notes that 16 cmplant grows 4 cm each day, and the 24 cm plant grows 3·5 cm each day.(a)Draw up a table showing the heights of the two pants each day for the week, starting onthe day that John got them.(b)Write down two formulas – one for each plant – to represent the plant’s height on anygiven day. State clearly the meaning of any letters used in your formulas.(c)John assumes that the plants will continue to grow at the same rates. Draw graphs torepresent the heights of the two plants over the first four weeks.(d)(i)(ii)(e)Check your answer to part (d)(i) using your formulae from part (b).(f)The point of intersection can be found either by reading the graph or by using algebra.State one advantage of finding it using algebra.(g)John’s model for the growth of the plants might not be correct. State one limitation ofthe model that might affect the point of intersection and its interpretation.7From your diagram, write down the point of intersection of the two graphs.Explain what the point of intersection means, with respect to the two plants. Youranswer should refer to the meaning of both co-ordinates.Project Maths Development Team 2012
EXTRA QUESTIONS1.17TICK TOCKEach hour, a clock chimes the number of times that corresponds to the time of day. Forexample, at three o’clock, it will chime 3 times. How many times does the clock chime in a day(24 hours)?1.18THE THEATREA theatre has 15 seats on the first row, 20 seats on the second row, 25 seats on the third row,and so on and has 24 rows of seats. How many seats are in the theatre?1.19“T TERMS”The T-shaped figure are in the table below is called T13 as it has 13 as initial )Find the value of T26 .(b)Find the sum of the numbers in the nth T in terms of n.(c)The set of numbers from which you can choose n.Modular Course 38
MATCH UP THE STORIES TO THE TABLES, GRAPH, AND FORMULAEStoryTableA cookbookrecommends 45minutes per kg tocook a turkey plusan additional 20minutes.kg123456John works for 12per hour. The graphand table show howmuch money heearns for the hourshe has worked.h012345Graphmins65110155200245290Linear but notproportionaly 20 45xThe graph is a line butit does not passthrough(0, 0)80 012243648606040y 12xRelationship is linearand proportionaly 100xRelationship is linearand proportional2000Changing euro tocent.euro012345cent0100200300400500 Salesman gets paid 400 per week plusan additional 50 forevery car sold.Carssold012345Henry has a winningticket for a lottery fora prize of 100. Theamount he receivesdepends on howmany others havewinning tickets.n1234569FormulaWhich of the followingapply? Linear, non linear,proportional or nonproportional? Justify246Relationship is linearbut not proportional400450500550600650y 400 50x 1005033.33252016.67y 100xCan you writethis formulaanother way?Project Maths Development TeamThe graph is a line butit does not passthrough(0, 0).Relationship is nonlinear and hence nonproportional.(In this case it isinversely proportional.) 2012
QUADRATIC AND CUBIC PATTERNS2.1MOTORBIKE STUNTUsing the information provided in the graph below;Find out if the graph is quadratic and give a reason for your answer.Modular Course 310
2.2GROWING SQUARES PATTERNDraw the next two patterns of growing squares.We wish to investigate how the number of tiles in each pattern is related to the side length ofeach square. Identify the independent and dependent variables.Complete the table below:Side of length ofeach squareNumber of tiles tocompete eachsquare11243945678910(a)How are the number of tiles used to make each square related to the side length of thesquare? Write the answer in words and symbols.(b)What difference do you notice about this formula and the formulae for linearrelationships?(c)Looking at the table, do you think is the relationship linear? Explain your answer.(d)Predict what a graph of this situation will look like. Will it be a straight line?Explain your answer. Plot a graph to check your prediction.(e)What shape is the graph? Does the rate of change of the number of tiles constant as theside length of each square increases? Explain using both the table and graph.(f)Why is the graph not a straight line? How can you recognise whether or not a graph willbe a straight line using a table?11Project Maths Development Team 2012
Let’s Investigate:We will redraw our first table, but we will put in some extra columns:Number of tiles toSide of length ofcompete eachDifference in Valueeach squaresquare(Change)112439Next Difference inValue(2nd Change)32545678910We saw that the changes in the table were not constant. Is there a pattern to them? Completethe table above and calculate the change of the changes.Can you see a pattern in the last two columns of the table above? What do you notice aboutthe last column (the 2nd change)When the change of the changes is constant we call this relationship between variables aquadratic relationship.Graph of growing squares:Does your graph look like this?List 3 properties (characteristics) of a quadraticrelationship which you have discovered from thisexercise.Modular Course 312
2.3GRAPHING & INVESTIGATINGPart 1: PERIMETERComplete a table for the perimeter of the base for cubes with different edge lengthsEdgeLength (cm )Perimeter ofthe base of thecube (cm)1237(a)Predict the shape of the graph.(b)Explain your prediction.(c)Write a formula for the perimeter of the base in terms of edge length.(d)Plot a graph to show the above relationship.(e)Check if values for perimeter predicted by the formula agree with values predicted bythe graph.13Project Maths Development Team 2012
Part 2: SURFACE AREAComplete the table below for total surface area of the cubeEdgeLength (cm)Surface Areaof thecube (cm2)11231 Unit1 Unit7(a)Predict the shape of graph for the above relationship.(b)Explain your prediction.(c)Write a formula for the total surface area in terms of the edge length.(d)Plot a graph to show the above relationship.Part 3: VOLUMEComplete the table below to find the volume of the cube.Volume of the cube: This refers to the total size, orhow much space the cube occupies.Volume Length x Width x HeightEdgeLength (cm)Volume of thecube (cm3)111st Change2nd Change3rd Change237Modular Course 314
(a)Predict the shape of graph for the above relationship.(b)Explain your prediction.(c)Write a formula for the volume of the cube in terms of the edge length.(d)Plot a graph to show the above relationship.(e)Check if values for volume predicted by the formula agree with values predicted by thegraph.RELATIONSHIP BETWEEN SURFACE AREA AND VOLUME FOR A CUBEDo you think the volume and surface area of a cube will ever be equal numerically?Do you think the volume will ever be numerically greater than the surface area?Check by completing the following table:Edge123456789nArea of BaseTotal surfaceAreaVolumeSurface AreaVolume(a)At what point is the ratio of surface area to volume equal to 1?(b)When is volume less than surface area?(c)When is volume greater than surface area?(d)How can you explain the rapid growth of volume and the slower growth of surfacearea?15Project Maths Development Team 2012
2.4ALGAE BLOOMWeek 0: 500m2Week 1: 700m2Week 2: 980m2Week 3: 1372m2(a)Is the pattern linear or quadratic? Explain your reasoning.(b)At what rate is the surface area covered by the Algae Bloom increasing?(c)Use a graph to extend the pattern to 8 weeks. Then make a scatter plot of the data anddescribe the graph.(d)The surface area of the lake is about 100,000 square metres. How many weeks does ittake the Algae Bloom to cover the entire lake?Extension Activity:Use a spreadsheet to extend the pattern to 20 weeks. Then make a scatter plot of the dataand describe the graph.Further information:One of the greatest difficulties facing the aquaculture industry in Ireland is the threat of harmfulalgae blooms. A bloom results in high populations of microscopic plant cells known asphytoplankton in the water. When a bloom dies off there can be depletion of oxygen in thewater column, and this kill in both fin-fish and shellfish populations, explains Mary Hensey, GlanUisce.Source: http://www.inshore-ireland.com/Modular Course 316
2.5 WHO GETS THE BETTER DEAL? You and your friend have both been offered a job on a construction site. You will both have to work 28 consecutive days to finish the project. Your friend is offered 25,000 per week (for 4 weeks) You negotiates your contact as follows; You will get paid 2 cent for the first day, 4 cent for the second day, 8 cent for the thirdday, and so on, your pay will double each day for 28 days.Who has negotiated the better deal?Create a table to show how much money you will get for the first10 days of the project.(a)Using the table would you expect a graph of this relationship to be linear? Explain.(b)Using the table would you expect a graph of this relationship to be quadratic? Explain.(c)Check changes, change of the changes, change of the change of the changes etc.(d)What do you notice? Is there a pattern to the differences? If so what is it?(e)Predict the type of graph you will get if you plot money in cent against time in days.(f)Make a graph to check your prediction. What do you notice?(g)Can you come up with a formula for the amount of money you will have after n days?(h)Using your calculator find out how much money you would get on the 5th day, the 10thday, the 20th day, 28th day?(i)What are the variables in the situation? What is constant in this situation?(j)Where is the factor of 2(the doubling) in the table? Where is it in the graph?(k)Contrast this situation with adding 2 cent every day. How much would you have on day31?(l)How would the formula change if your pay trebled each day, starting by giving you 2cents on the first day, 6 cent on the second day, 18 cents on the third day, and so on asdescribed above? Make a table and come up with the formula for this new situation.(m) When people speak of ‘exponential growth’ in everyday terms, what key idea are theytrying to communicate?17Project Maths Development Team 2012
2.6IDENTIFYING GRAPHSBelow are 4 sections of 4 different graphs:You must decide which one is a linear, a quadratic, a cubic and an exponential is using theinformation provided.In each case, give a reason for your answer.2.7GENERAL FORMULA 1Write the general formula for the following patterns.(a)(b)2.83,12,27,48,75, 0.25,1,2.25,4,6.25, GENERAL FORMULA 2Write the general formula for the following patterns.(a)5,12,21,32,45, (b)5,15,31,53,81, 2.9MORE PATTERNS(a)How many blocks are in the 4th pattern?(b)Write a general formula to find the number of in the nth pattern.(c)How many blocks are in the 8th pattern?Modular Course 318
2.10GEOMETRIC SEQUENCES 1Given the sequence 3 12 48 (a)(b)(c)Show that it is GeometricFind the Tn, Sn of the sequenceFind the T10 , S10 of the sequence2.10GEOMETRIC SEQUENCES 2The five numbers 48, p, q, t, 3 are in a Geometric Sequence.Find p, q and t.2.11GEOMETRIC SEQUENCES 3A ball is dropped from a height of 8 m. The ball bounces to 80% of its previous height with eachbounce. How high (to the nearest cm) does the ball bounce on the fifth bounce.19Project Maths Development Team 2012
FINANCIAL APPLICATIONS TO SEQUENCES AND SERIES3.1THE SNOWMAN [JCHL 2007 PAPER 1]A snowman has a mass of 12 kg. It melts at a rate of 0.2% of its mass per minute.What will be the mass of the snowman after 3 minutes?Give your answer correct to 2 decimal places.3.2DEPOSIT IN THE BANK 650 is deposited in a fixed interest rate bank account. The amount in the account at the endof each year is shown in the following table:End of yearFinal value1 676.002 703.043 731.164 760.415 790.82(a)How can you tell from the table if the above relationship is linear, quadratic orexponential? Explain.(b)If you plot a graph of final value against time what does the graph look like for thislimited range of times?(c)What formula expresses the final value after t years if this pattern continues given aninitial value of 650?(d)What would be the effect of increasing the interest rate? Make a table showing the finalvalues for the first five years using an interest rate of 10% per annum compound interest.Plot a graph for this data. Compare the graph to the graph produced for the lowerinterest rate.3.3NEW JOBJoan gets a new job as a trainee. She starts on 40 per day. She is told that in 6 months she willget a 50% rise and in another 6 months she will get another 50% rise. She says “Great, in oneyear’s time I will have doubled my money.” Discuss.3.4TECHNOLOGY INNOVATIONSAn item was being produced for 16 twenty years ago. Due to technology innovations it wasreduced by 50% ten years ago and reduced again by 50% recently. Is the item now free?Discuss.3.5MONEY IN THE BANKJohn put 200 into the bank for 1 year and got 10% interest during that year. At the end of theyear he had 220. This means that he had gained 20 on his original money.Task: Match John’s figures to each of the words in the table below:PrincipalInterest rateas apercentageInterest rateas a decimalFinal ValueModular Course 3Number ofyearsInterest20
3.6SAVINGSThe following figures represent a certain amount of money put into a bank for a certain numberof years at a certain interest rate. Using all of the words in the table above, write out a fewsentences which would explain all of the figures.Figures:3.7 472.054 years 30012%0.12 172.05INVESTMENTSThe table below shows money invested by various people for a differing number of years. Someof the figures are missing. Complete the missing figures:Note: p.a. means per annum (per year)Interest rateNumber ofNamePrincipalFinal ValueInterest% (p.a.)yearsAnne 1 0006%Michael 1 0007%Dominic218% 1338.23 5 038.85Project Maths Development Team59 838.463 1 038.85 2012
3.8CONVERSIONSBelow are some annual interest ratesexpressed as decimals.Convert these to annual rates expressed aspercentage rates.Below are some annual interest ratesexpressed as percentages.Convert these to annual rates expressed .35%0.002750.0246%0.000350.00035%This is the “i” in the“Formulae and Tables”book.3.9PATRICK’S INVESTMENTPatrick invests 400 in a savings account for 1 year and gets a fixed annual equivalent rate(AER) of 4.3%. At the end of the year he asks the bank how much money he has in total andhow much interest he made. Fill out the table below to see what figures the bank might leavehim.Method 1PrincipalInterest for the year(calculate 4.3% of 400)Final ValueModular Course 322
3.10MARY’S GIFTMary has received a gift of 5000. She is hoping to buy a car for 6 000 with her savings in 3years’ time. She intends to save the gift money until then. The bank is offering her 4% AER on hermoney if she leaves it in for the 3 years. Will she be able to afford to buy the car from hersavings at the end of the three years?Method 1Value of the gift (P)Interest for the 1st year (I1)(4% of 5 000)F1 Final value (end of year 1)Interest for the 2nd year (I2)F2 Final Value (end of year 2)Interest for the 3rd year (I3)F3 Final Value (end of year 3)I1Fill in the table on the right.What do you notice?3.11I2I3JOHN’S SAVINGSJohn wants to have 10 000 saved in 10 years’ time to pay for his child’s education. The bank isoffering him an annual interest rate of 7% AER. How much money would he need to invest nowin order to have 10 000 in 10 years’ time?First fill in as many variables as you can into the table below:P3.12F(1 i)tFIONA’S SAVINGSFiona has 7,000 to put into a savings account. She would like 10 000 in 4 years’ time in order tobuild an extension to her house. She decides to ask a few banks, building societies etc. whatrate of interest they are willing to offer. What annual rate of interest does Fiona need to haveenough saved to build the extension?First fill in as many variables as you can into the table below and then calculate i:P23i(1 i)Project Maths Development TeamtF 2012
3.13GAME CONSOLESJillian and Noel are each going to buy games console. It costs 500 and they are getting aloan from the credit union to do it. Jillian says “I have loads of work at present so I can afford topay 100 per month.” Noel says he can only afford 80 per month. The bank is charging them amonthly interest rate of 1%.(a)Complete the first three months’ transactions on the loan using the boxes below.Jillian’s First 3 MonthsNoel’s first 3 monthsInitial LoanInitial LoanInterest 1Interest 1TotalTotal(500 x 0.01)(500 x 0.01)PaymentPaymentBalanceBalanceInterest 2Interest 2TotalTotalPayment 2Payment 2Balance 2Balance 2Interest 3Interest 3TotalTotalPayment 3Payment 3Balance 3Balance 3(b)Compare: Interest 1, Interest 2 and Interest 3 for Jillian or Noel. What do you notice?Explain.(c)Explain what is meant by the term “Reducing Balance in the context of either Jillian’s orNoel’s situation ?(d)Investigate the effects of their loan repayments under two headings Time taken to repay Interest paid.Modular Course 324
3.14SALE OF LORRYA company buys a new lorry for 50 000. After 4 years it needs to sell the lorry. The value of thelorry reduces by 15% each year. What is the value of the lorry after 4 years?Method 1Original value of the lorry PDepreciation in year 1 50000(0.15)F1 Value at end of year 1Depreciation in year 2F2 Value at end of year 2Depreciation in year 3F3 Value at end of year 33.15LCOL NCCA 2011A machine depreciates in value by 40% in its first year of use. During its second year itdepreciates by 25% of its value at the beginning of that year. Thereafter, for each year, itdepreciates by 10% of its value at the beginning of the year.Calculate:(i)the value after eight years of equipment costing 500 new(ii)the value when new of equipment valued at 100 after five years of use.3.16LEAVING CERTIFICATE 2010 SAMPLE PAPER 1 FOUNDATION LEVEL Q2A sum of 5000 is invested in an eight year government bond with an annual equivalent rate(AER) of 6%.Find the value of the investment when it matures in eight years’ time.3.17LEAVING CERTIFICATE 2010 SAMPLE PAPER 1 ORDINARY LEVEL Q2(a)A sum of 5000 ia invested in an eight-year government bond with an annual equivalentrate (AER) of 6%. Find the value of the investment when it matures in eight years’ time.(b)A different investment bond gives 20% interest after 8 years. Calculate the AER for thisbond.3.18COMPOUND INTEREST 1 100 earns 0.287% per month compound interest.(i)What is its final value after 1 year?(ii)If interest was added annually what is the annual equivalent rate?25Project Maths Development Team 2012
3.19COMPOUND INTEREST 2The 100 is left on deposit for 15 months at 0.287% per month compound interest.(i)Calculate the final value. Give the answer to the nearest 10 c.(ii)What is the interest rate for the 15 months?(iii)What is this interest rate called?3.20AER ADVERTIMENTVerify the following AER:3.21COMPOUND INTEREST 3(i)Find, correct to three significant figures, the rate of interest per month that would, if paidand compounded monthly, be equivalent to an effective annual rate of 3.5%?(ii)Find, correct to three significant figures, the rate of interest per day that would, if paidand compounded daily, be equivalent to an effective annual rate of 3.5%?3.22FUNDSA bank has offered a 9 month fixed term reward account paying 2.55% on maturity, for newfunds from 10,000 to 500,000. (That is, you get your money back in 9 months’ time, along with2.55% interest.)Confirm that this is, as advertised, an EAR of 3.4%Modular Course 326
1.10 DESIGNER TILES Caroline bought 400 designer tiles at an end of line sale. She wants to use them as a feature in her new bathroom. . The point of intersection can be found either by reading the graph or by using algebra. State one advantage of finding it using algebra. (g) John’s mo
