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Ib math hl portfolio ia the dice game/filling up the petrol tank/shadow functions/patterns from complex numbers help tutors sample example solution guide

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Now Ex-IB Examiner's Online or face to face Help: +91 9911918255 or +91 9716570693 Email: ramprhr@gmail.com Website: http://www.topclassibtutors.com/ Online Help/Tuition all over the world; Mail us or ...

Now Ex-IB Examiner's Online or face to face Help: +91 9911918255 or +91 9716570693 Email: ramprhr@gmail.com Website: http://www.topclassibtutors.com/ Online Help/Tuition all over the world; Mail us or call us for any queries:

MATHEMATICS
Higher Level
The portfolio – IA tasks
For use in 2012 and 2013
SHADOW FUNCTIONS HL TYPE I

While real zeros of polynomial functions may be easily read off the graph of the
polynomial, the same is not true for complex zeros. In this task, you will investigate the
method of shadow functions and their generators, which helps identify the real and
imaginary components of complex zeros from key points along the x-axis.
Part A (Quadratic Polynomials)
Consider the quadratic function 2 2
Write down the coordinates of the vertex.
Show that 1 y has zeros, where.

_ How can the findings with quadratics and cubics be applied to quartics?
PATTERNS FROM COMPLEX NUMBERS HL TYPE I
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
and answer” format.

THE DICE GAME HL Type II
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
and answer” format.
The aim of this task is to create a dice game in a casino and model this using
probability. It is important to examine how best to run the game from both the
perspective of a player and the casino. In doing so analyse the game to consider the
optimal payments by the player and payouts by the casino.
_ Consider a game with two players, Ann and Bob. Ann has a red die and Bob a white die.

_ Now consider other models for the game including cases such as where the player or the
bank rolls their dice multiple times, or where multiple players are involved in the game.
FILLING UP THE PETROL TANK HL Type II
To the student: The work that you produce to address the questions in this task should be a report
that can stand on its own. It is best to avoid copying the questions in the task to adopt a “question
and answer” format.
In this task, you will develop a mathematical model that helps motorists decide which of
the following two options is more economical.
Option 1: to buy petrol from a station on their normal route at a relatively higher price.
Option 2: to drive an extra distance out of their normal route to buy cheaper petrol.
Consider a situation in which two motorists, Arwa and Bao, share the same driving route but own
different sized vehicles. Arwa fills up her vehicle’s tank at a station along her normal route for
US$ 1 p per litre. On the other hand, Bao drives an extra d kilometres out of his normal route to fill
up his vehicle’s tank for US$ p2 per litre where p2 _ p1 .
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    Ib math hl portfolio ia the dice game/filling up the petrol tank/shadow functions/patterns from complex numbers help tutors sample example solution guide Ib math hl portfolio ia the dice game/filling up the petrol tank/shadow functions/patterns from complex numbers help tutors sample example solution guide Document Transcript

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    • The following diagram shows a circle with centre O and radius r, and any point P. 1 CThe circle has centre P and radius OP. Let A be one of the points of intersection of and . Circle has centre A, and radius r. The point is the intersection of with (OP). This is shown in the diagram below.Let . Use an analytic approach to find , when , and . Describewhat you notice and write a general statement to represent this.Let . Find , when , and . Describe what you notice and write ageneral statement to represent this. Comment whether or not thisstatement is consistent with your earlier statement.Use technology to investigate other values of r and OP. Find thegeneral statement for . OPTest the validity of your general statement by using differentvalues of OP and r.Discuss the scope and/or limitations of the general statement. Explain how you arrived at the general statement.LACSAP’S FRACTIONS SL TYPE IAim: In this task you will consider a set of numbers that arepresented in a symmetrical pattern. Consider the five rows of numbers shown below.Describe how to find the numerator of the sixth row.Using technology, plot the relation between the row number, n,and the numerator in each row. Describe what you notice fromyour plot and write a general statement to represent this.Find the sixth and seventh rows. Describe any patterns you used.
    • Let be the element in the nth row, starting with . Example: .Find the general statement for.Test the validity of the general statement by finding additionalrows.Discuss the scope and/or limitations of the general statement.Explain how you arrived at your general statement.FISH PRODUCTION SL TYPE IIAim: This task considers commercial fishing in a particularcountry in two different environments – the sea and fish farms(aquaculture). The data is taken from the UN Statistics DivisionCommon Database.The following table gives the total mass of fish caught in thesea, in thousands of tonnes (1 tonne = 1000 kilograms).Define suitable variables and discuss any parameters/constraints.Using technology, plot the data points from the table on a graph.Comment on any apparent trends in your graph and suggestsuitable models.Analytically develop a model that fits the data points. (You mayfind it useful to consider a combination of functions.)On a new set of axes, draw your model function and the originaldata points. Comment on any differences. Revise your model ifnecessary.The table below gives the total mass of fish, in thousands oftonnes, from fish farms.
    • Plot the data points from this table on a graph, and discusswhether your analytical model for the original data fits the newdata.Use technology to find a suitable model for the new data. On anew set of axes, draw both models.Discuss how trends in the first model could be explained bytrends in the second model.By considering both models, discuss possible future trends inboth types of fishing.GOLD MEDAL HEIGHTS SL TYPE IIAim: The aim of this task is to consider the winning height forthe men’s high jump in the Olympic Games.The table below gives the height (in centimeters) achieved bythe gold medalists at various Olympic Games.Note: The Olympic Games were not held in 1940 and 1944.Using technology, plot the data points on a graph. Define allvariables used and state any parameters clearly. Discuss anypossible constraints of the task.What type of function models the behaviour of the graph?Explain why you chose this function. Analytically create anequation to model the data in the above table.On a new set of axes, draw your model function and the originalgraph. Comment on any differences. Discuss the limitations ofyour model. Refine your model if necessary.Use technology to find another function that models the data. Ona new set of axes, draw both your model functions. Comment onany differences.
    • Had the Games been held in 1940 and 1944, estimate what thewinning heights would have been and justify your answers.Use your model to predict the winning height in 1984 and in2016. Comment on your answers.The following table gives the winning heights for all the otherOlympic Games since 1896.How well does your model fit the additional data?Discuss the overall trend from 1896 to 2008, with specificreferences to significant fluctuations.What modifications, if any, need to be made to your model to fitthe new data?MATHEMATICSHigher LevelThe portfolio – IA tasksFor use in 2012 and 2013SHADOW FUNCTIONS HL TYPE ITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.While real zeros of polynomial functions may be easily read offthe graph of thepolynomial, the same is not true for complex zeros. In this task,you will investigate the
    • method of shadow functions and their generators, which helpsidentify the real andimaginary components of complex zeros from key points alongthe x-axis.Part A (Quadratic Polynomials)Consider the quadratic function 2 2Write down the coordinates of the vertex.Show that 1 y has zeros, where.The “shadow function” to 1 y is another quadratic 2 y whichshares the same vertex as.However, 2 y has opposite concavity to that of 1 y and its zerosare in the form.Use various values for a and b to generate pairs of functions 1 yand.Hence, or otherwise, express 2 y in terms of 1 y and m y , wherem y is called the “shadowgenerating function”._ On a labelled diagram, illustrate how the zeros of 2 y may behelpful in the determinationof the real and imaginary components of the complex zeros of 1y.Part B (Cubic Polynomials)Now consider the cubic function 1 _ _ _ _ y _ (x _ 2) x _ (3_2i) x _ (3_ 2i) .The shadow function in this case is another cubic function 2 ywhich shares two points with 1 y ,has opposite concavity and its zeros are _2 and 3_ 2 as depictedin the figure below. The shadow
    • generating function in this case passes through the two pointsof intersection.Write down an expression for 2 y and hence find the points ofintersection between 1 y and 2 y ._ Hence, or otherwise, determine the equation of the shadowgenerating function m y in thiscase._ Once again, express 2 y in terms of 1 y and m y . Usetechnology to investigate similarcubic functions._ Write a general statement about the functions 1 y , 2 y and m yand prove your statement._ On a labelled diagram, illustrate how the zeros of 2 y may behelpful in the determinationof the real and imaginary components of the complex zeros of 1y._ How can the findings with quadratics and cubics be applied toquartics?PATTERNS FROM COMPLEX NUMBERS HL TYPE ITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.Part AUse de Moivre’s theorem to obtain solutions to the equation z3_1 _ 0 .
    • _ Use graphing software to plot these roots on an Arganddiagram as well as a unit circlewith centre origin._ Choose a root and draw line segments from this root to theother two roots._ Measure these line segments and comment on your results._ Repeat the above for the equations z4 _1 _ 0 and z5 _1 _ 0 .Comment on your results andtry to formulate a conjecture._ Factorize zn _1 for n _ 3, 4 and 5._ Use graphing software to test your conjecture for some morevalues of n � __ and makemodifications to your conjecture if necessary._ Prove your conjecture.Part BUse de Moivre’s theorem to obtain solutions to zn _ i for n _ 3,4 and 5._ Use graphing software to represent each of these solutions onan Argand diagram._ Generalize and prove your results for zn _ a _ bi , where a _bi _1._ What happens when a _ bi _1?THE DICE GAME HL Type IITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.
    • The aim of this task is to create a dice game in a casino andmodel this usingprobability. It is important to examine how best to run the gamefrom both theperspective of a player and the casino. In doing so analyse thegame to consider theoptimal payments by the player and payouts by the casino._ Consider a game with two players, Ann and Bob. Ann has ared die and Bob a white die.They roll their dice and note the number on the upper face. Annwins if her score is higherthan Bob’s (note that Bob wins if the scores are the same). Ifboth players roll their diceonce each what is the probability that Ann will win the game?_ Now consider the same game where Ann can roll her die asecond time and will note thehigher score of the two rolls but Bob rolls only once. In this casewhat is the probabilitythat Ann will win?_ Investigate the game when both players can roll their dicetwice, and also when bothplayers can roll their dice more than twice, but not necessarilythe same number of times._ Consider the game in a casino where the player has a red dieand the bank has a white die.Find a model for a game so that the casino makes a reasonableprofit in the case where theplayer rolls the red die once and the bank rolls the white dieonce. (When creating yourmodel you will need to consider how much a player must pay toplay a game and how
    • much the bank will pay out if the player wins. Do this from theperspective of both theplayer and the casino and consider carefully the criteria forwhether the game can beconsidered worthwhile for both the player and the casino.)_ Now consider other models for the game including cases suchas where the player or thebank rolls their dice multiple times, or where multiple playersare involved in the game.FILLING UP THE PETROL TANK HL Type IITo the student: The work that you produce to address thequestions in this task should be a reportthat can stand on its own. It is best to avoid copying thequestions in the task to adopt a “questionand answer” format.In this task, you will develop a mathematical model that helpsmotorists decide which ofthe following two options is more economical.Option 1: to buy petrol from a station on their normal route at arelatively higher price.Option 2: to drive an extra distance out of their normal route tobuy cheaper petrol.Consider a situation in which two motorists, Arwa and Bao,share the same driving route but owndifferent sized vehicles. Arwa fills up her vehicle’s tank at astation along her normal route forUS$ 1 p per litre. On the other hand, Bao drives an extra dkilometres out of his normal route to fillup his vehicle’s tank for US$ p2 per litre where p2 _ p1 .
    • _ Choose suitable parameters for Arwa’s and Bao’s vehicles.Justify your choices._ Suppose 1 p _ US$1.00 per litre, 2 p _ US$0.98 per litre and d_10 km. Which motorist is getting the better deal? Show allcalculations and justify any assumptions you make._ On a spreadsheet, choose several sets of values for 1 p , 2 pand d to investigate further._ Define a set of variables that would be relevant in the abovesituation.“Effective litres” is a method of comparing the cost of petrolbought under the two optionsdescribed above. Effective litres, for a given vehicle, are thoselitres used when the vehicle travelsits normal route._ Use your variables and parameters to write algebraicexpressions for 1 E and 2 E whichrepresent the cost per effective litre under options 1 and 2respectively._ Write a model that helps motorists decide on the moreeconomical option for theirvehicles.In the remainder of this task, you need to consider the twovehicles you have chosen for Arwaand Bao._ Use your model to find the farthest distance that Bao shoulddrive to obtain a 2 % pricesaving._ Investigate the relationship between d and 2 p when 2 E iskept constant (e.g. US$0.90,US$1.00, … etc.). Use technology to draw a family of curves forArwa’s vehicle. Repeat
    • for Bao’s vehicle._ For 2 E _ US$0.90 , provide Arwa with information on threedifferent stations that yieldthis same cost per effective litre. Discuss how such informationmay be useful to Arwa._ For 2 E _ US$1.00 and 2 p _ US$0.80 , compare themaximum distance that each motoristshould drive and still save money.Arwa is a busy person and wonders whether the saving inmoney would be worth the time shewould lose in extra driving._ Modify your model to account for the time taken to drive toand from an off-route station.Clearly justify any assumptions you make.IB DP Math SL type 2 IA portfolio task Fish production, GoldMedal heightsIB DP Math HL type 2 IA portfolio task The dice game, Fillingup the petrol tank.IB DP Math HL type 1 IA portfolio task shadow function,Patterns from complex number.IB DP Math SL type 1 IA portfolio task Lacsap’s fraction,Circles IB ToK Essay Tutor IB ToK Essay Topics 2012/13 IB ToK Essay Help: Choosing the Right Essay Title IB ToK Essay Help: Planning and Writing the Essay IB ToK Essay Help: Marking and Assessing the Essay
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    • IB, Maths,Portfolio, urdu shayari help,HL/SL,complex ,number,dice game,fish ,production,Ib, filling up,petrol tank, shadow function,numbers,gold medal heights,Lacasp’s fraction,circles,tutorIB math home tuition in delhi IB maths home tuition in delhi IB maths home tuition in delhi IB math home tuition in delhi IB math home tuition in gurgaon IB maths home tuition ingurgoan IB maths home tuition in gurgoan IB math home tuition in gurgaon IB Mathematics Physics Chemistry Economics Business and Management HL SL Studies Theatre Stimulus help IB Physics Mathematics Chemistry Economics Business and Management Presentation ITGS IA Computer help IB Chemistry HL SL help Mathematics Physics Economics Chemistry Business IB Economics HL SL Management Math Portfolio TOK Economics Historical Investigation Extended help IB Business and Management HL SL help Essay Commentary storical Investigation Lab writing IB Math Portfolio Help HL SL help Lab writing Project ESS English Spanish French Hindi IB Math Project Help Studies help IB World Literature Theatre Stimulus Essay Commentary storical Investigation IB Extended Essay Help TOK Presentation ITGS IA Computer Dossier IB World Literature Theatre StimulusIB TOK Essay Help Mathematics Physics Economics Chemistry Business Presentation ITGS IA Computer IB Economics Commentary Help Management Math Portfolio HL SL Extended IB Historical Investigation Help Essay Commentary storical Investigation Lab writing
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    • Call or Email Ex-IB Examiner for Help: +91 9911918255 or +91 9716570693/ Email: ramprhr@gmail.com Website: http://www.topclassibtutors.com/ Online Help/Tuition allover the world by the Best/Expert IB Tutors; Mail us or call us for any queriesEssential Elements of a Math Portfolio Internal Assessment (IA)Task hl/sl type 1 type 2 Problem Write--Up General-­­Name, class, and due date are clearly displayed.Introduction--Write up has an interesting (or pertinent) title.-­­Task is properly introduced to the reader.-­­This introduction should pose any major questions the task isseeking to answer.Body-­­Problem-­­solving process is completely explained throughout
    • -­­Write-­­up is written in a way that would allow a youngerstudent to completely understand the problem posed and thereasoning that lead to the solution.-­­Write-­­up is written as a technical paper, not a piece of prose.-­­Paragraphs are concise and to the point.-­­Use of a variety of technology is regularly used and cited.-­­Graphs and pictures are inserted in meaningful and organizedways.-­­Graphs and pictures are completely visible, properly cited, andlabeled in appropriate and meaningful ways.-­­All equations are displayed using anequation-­­writing program.
    • -­­Questions are answered in a logical order withoutnecessarily restating each question.-­­All answers are correct and well supported by mathematicalevidence or reasoning that is obvious to the reader.Conclusion-­­All questions posed throughout the task are thoroughlyanswered.-­­A general form of a solution has been reached andexplained.-­­Investigation beyond the stated task has beenattempted.
    • Basic Elements of a Good IB Mathematics Portfolio/ IB MathInternal Assessment (IA) Task HL/SL Type 2 Type 1 ProjectCorrect answers throughout.All questions answered in a logical order.Audience is the teacher.Identify and address the principal goal of the assignment.Written explanations of difficult or less obvious steps in work.Word-processed including use of technology for graphs andequations.Elements of Superior WorkAttention to detail.Consistent use of significant figures or decimal place rounding.Use of proper notation such as “≈” or approximations.Avoid calculator notation such as 4*5, x^2, or 2.3E4.Graphs are clear with logical scales and labeled axes.Clearly define all variable that are introducedWork stands alone.Audience is another student of the same level.
    • Reader doesn’t have to refer to the assignment sheet.Reads smoothly and effortlessly.Answers are not numbered unless required.Single word-processed documentIntegrate an equation-writing program, spreadsheets, graphingprogram, etc.Comment on everything included.Explain all steps each time a new mathematical process isused.Avoid repetition of processes already explained.Graphs, charts, etc. appear on the same page where they areexplained.Student goes beyond the assignment with insight andsophistication.Identify and address the hidden goals of the assignment.Try to get into the mind of the writer of the assignment.Most tasks leave some space for the top students to explore.Student uses a variety of mathematical approaches.A good assignment will require the use of many kinds of math.
    • A good question will often have several possible approachesfor the solution.All math used is relevant to the task.No fluff or space-fillers.Tips to the StudentsWork through all problems with pencil and calculator beforewriting it up. Gives an overall feel for the project so that that student canstart to address the main goal, andeven insights, right from the start.Write a brief introduction rather than jumping right in to thefirst question.Sets a tone of readability.Math SL/HL students can pretend that they are writing achapter in a textbook. Encourages students to fully explain each step, rememberingthat their audience is anotherstudent from their course of average ability.Students can have fun with creative formatting.Take pride in the look and feel of their work.
    • Math Studies students can approach the IA Project as theywould a science lab.Write a hypothesis.Collect data.Analyze data.Draw conclusions.Discuss possible errors or improvements.Common ProblemsSolutions to individual problems are cut in half by the pagebreak.It is ok to leave space on the bottom of a page.Use the “Insert Page-Break” feature of most word-processingprograms when a new problem isstarted. (If there is not enough space on the current page.)Proof for comments like “see the graph below” when the graphwas accidentally bumped to thenext page during editing.Pages of graphs or charts with comments and explanations atend.Full-page graphs should be avoided if possible.
    • Include small (but clear) graphs with an explanation on thesame page. Use features like “text-wrapping” to make graphs or chartslook and feel like an integrated partof the work.If a graph must take up a full page, include a full discussion of iton the next page.Pages of raw data in the main body of text. Raw data should be organized as concisely as possible andincluded in the appendix. The main body should only include the compiled data used inthe relevant calculation.Frequent reference to appendix.Cuts the flow of the work.If what is in the appendix is very important, move it to thebody.Over-researching.IB Math Internal Assessments are not research projects.IB is looking for student work, and it is obvious when it is not.Grading
    • Read paper from start to finish before assigning grades.Detail is important.Overall feel is important.Make comments or corrections in the text.Comments help the IB moderator to understand how you haveassigned marks.Reference the IB Rubric on a regular basis IB has two rubrics which could be used for students at manylevels.Math Studies. (Open-ended project.)Math SL/HL. (Answering an assigned problem.)We are expert ib math tutors who have been helping ib students across the world through online. Wedo all types of ib mathematics ia/portfolio type 2 type 1 hl and sl. Here are some tips on how to do mathia circles sl type 1Always begin with Good impressive introduction, predict what you are going to do in whole ia.Mention what technological tool you are going to use.If asked about model then dont forget to validate your model, a model without validation has nomeaning.Put Proper Heading, Figure Caption, well labelled graph, and Table heading followed by someexplanation what is happening.For further help, you can contact us, we give 100% guarantee for top class grades. Top class IB tutors,top class results.Please feel free to contact us for any queries or help on ib math ia circles sl type 1 portfolio write up ibtutors sample example solution guidance; mail us or call us: +91 9911918255 or +919716570693
    • Email: ramprhr@gmail.com/ Website: www.topclassibtutors.comNow, Ex-IB Examiner for online help/ Tuition across the world