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MATHEMATICS Standard Level The portfolio – IA tasks For use in 2012 and 2013 CIRCLES SL TYPE I Aim: The aim of this task is to investigate positions of points in intersecting circles. The following diagram shows a circle with centre O and radius r, and any point P. 1 C The 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 . Describe what you notice and write a general statement to represent this. Let . Find , when , and . Describe what you notice and write a general statement to represent this. Comment whether or not this statement is consistent with your earlier statement. Use technology to investigate other values of r and OP. Find the Test the validity of your general statement by using different values 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 I Aim: In this task you will consider a set of numbers that are presented 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 from your 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 additional rows. Discuss the scope and/or limitations of the general statement. Explain how you arrived at your general statement. FISH PRODUCTION SL TYPE II Aim: This task considers commercial fishing in a particular country in two different environments – the sea and fish farms (aquaculture). The data is taken from the UN Statistics Division Common Database. The following table gives the total mass of fish caught in the sea, 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 suggest suitable models. Analytically develop a model that fits the data points. (You may find it useful to consider a combination of functions.) On a new set of axes, draw your model function and the original data points. Comment on any differences. Revise your model if necessary. The table below gives the total mass of fish, in thousands of tonnes, from fish farms. Plot the data points from this table on a graph, and discuss whether your analytical model for the original data fits the new data. Use technology to find a suitable model for the new data. On a new set of axes, draw both models. Discuss how trends in the first model could be explained by trends in the second model. By considering both models, discuss possible future trends in both types of fishing. GOLD MEDAL HEIGHTS SL TYPE II Aim: The aim of this task is to consider the winning height for the men’s high jump in the Olympic Games. The table below gives the height (in centimeters) achieved by the 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 all variables used and state any parameters clearly. Discuss any possible constraints of the task. What type of function models the behaviour of the graph? Explain why you chose this function. Analytically create an equation to model the data in the above table. On a new set of axes, draw your model function and the original graph. Comment on any differences. Discuss the limitations of your model. Refine your model if necessary. Use technology to find another function that models the data. On a new set of axes, draw both your model functions. Comment on any differences. Had the Games been held in 1940 and 1944, estimate what the winning heights would have been and justify your answers. Use your model to predict the winning height in 1984 and in 2016. Comment on your answers. The following table gives the winning heights for all the other Olympic Games since 1896. How well does your model fit the additional data? Discuss the overall trend from 1896 to 2008, with specific references to significant fluctuations. What modifications, if any, need to be made to your model to fit the new data? MATHEMATICS Higher Level The portfolio – IA tasks For use in 2012 and 2013 SHADOW FUNCTIONS 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 qu stions in th task to adopt a “qu stion and answ r” format. 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. The “shadow function” to 1 y is another quadratic 2 y which shares the same vertex as. However, 2 y has opposite concavity to that of 1 y and its zeros are in the form. Use various values for a and b to generate pairs of functions 1 y and. Hence, or otherwise, express 2 y in terms of 1 y and m y , where m y is call d th “shadow g n rating function”. _ On a labelled diagram, illustrate how the zeros of 2 y may be helpful in the determination of the real and imaginary components of the complex zeros of 1 y. 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 y which shares two points with 1 y , has opposite concavity and its zeros are _2 and 3_ 2 as depicted in the figure below. The shadow generating function in this case passes through the two points of intersection. Write down an expression for 2 y and hence find the points of intersection between 1 y and 2 y . _ Hence, or otherwise, determine the equation of the shadow generating function m y in this case. _ Once again, express 2 y in terms of 1 y and m y . Use technology to investigate similar cubic functions. _ Write a general statement about the functions 1 y , 2 y and m y and prove your statement. _ On a labelled diagram, illustrate how the zeros of 2 y may be helpful in the determination of the real and imaginary components of the complex zeros of 1 y. _ 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 qu stions in th task to adopt a “qu stion and answ r” format. Part A Us d Moi r ’s th or m to obtain solutions to th quation z3 _1 _ 0 . _ Use graphing software to plot these roots on an Argand diagram as well as a unit circle with centre origin. _ Choose a root and draw line segments from this root to the other 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 and try to formulate a conjecture. _ Factorize zn _1 for n _ 3, 4 and 5. _ Use graphing software to test your conjecture for some more values of n �� __ and make modifications to your conjecture if necessary. _ Prove your conjecture. Part B Us d Moi r ’s th or m to obtain solutions to zn _ i for n _ 3, 4 and 5. _ Use graphing software to represent each of these solutions on an 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 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 qu stions in th task to adopt a “qu stion and answ r” 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. They roll their dice and note the number on the upper face. Ann wins if her score is higher than Bob’s (not that Bob wins if th scor s ar th sam ). If both players roll their dice once each what is the probability that Ann will win the game? _ Now consider the same game where Ann can roll her die a second time and will note the higher score of the two rolls but Bob rolls only once. In this case what is the probability that Ann will win? _ Investigate the game when both players can roll their dice twice, and also when both players can roll their dice more than twice, but not necessarily the same number of times. _ Consider the game in a casino where the player has a red die and the bank has a white die. Find a model for a game so that the casino makes a reasonable profit in the case where the player rolls the red die once and the bank rolls the white die once. (When creating your model you will need to consider how much a player must pay to play a game and how much the bank will pay out if the player wins. Do this from the perspective of both the player and the casino and consider carefully the criteria for whether the game can be considered worthwhile for both the player and the casino.) _ 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 qu stions in th task to adopt a “qu stion and answ r” 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 diff r nt siz d hicl s. Arwa fills up h r hicl ’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 hicl ’s tank for US$ p2 per litre where p2 _ p1 . _ Choos suitabl param t rs for Arwa’s and Bao’s hicl s. 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 all calculations and justify any assumptions you make. _ On a spreadsheet, choose several sets of values for 1 p , 2 p and d to investigate further. _ Define a set of variables that would be relevant in the above situation. “Eff cti litr s” is a m thod of comparing th cost of p trol bought under the two options described above. Effective litres, for a given vehicle, are those litres used when the vehicle travels its normal route. _ Use your variables and parameters to write algebraic expressions for 1 E and 2 E which represent the cost per effective litre under options 1 and 2 respectively. _ Write a model that helps motorists decide on the more economical option for their vehicles. In the remainder of this task, you need to consider the two vehicles you have chosen for Arwa and Bao. _ Use your model to find the farthest distance that Bao should drive to obtain a 2 % price saving. _ Investigate the relationship between d and 2 p when 2 E is kept constant (e.g. US$0.90, US$1.00, … etc.). Use technology to draw a family of curves for Arwa’s hicl . R p at for Bao’s hicl . _ For 2 E _ US$0.90 , provide Arwa with information on three different stations that yield this same cost per effective litre. Discuss how such information may be useful to Arwa. _ For 2 E _ US$1.00 and 2 p _ US$0.80 , compare the maximum distance that each motorist should drive and still save money. Arwa is a busy person and wonders whether the saving in money would be worth the time she would lose in extra driving. _ Modify your model to account for the time taken to drive to and from an off-route station. Clearly justify any assumptions you make. IB DP Math SL type 2 IA portfolio task Fish production, Gold Medal heights IB DP Math HL type 2 IA portfolio task The dice game, Filling up the petrol tank. IB DP Math HL type 1 IA portfolio task shadow function, Patterns from complex number. IB DP Math SL typ 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 IB ToK Essay Help: Grammar and Punctuation Check IB TOK Essay on Common Sense IB Sample TOK essays Examiners mark essays against the title as set. Respond to the title exactly as given; do not alter it in any way. Your essay must be between 1200 and 1600 words in length, double spaced and typed in size 12 font. 1. In what ways may disagreement aid the pursuit of knowledge in the natural and human sciences? 2. “Only s ing g neral patterns can give us knowledge. Only s ing particular xampl s can gi us und rstanding.” To what extent do you agree with these assertions? 3. “Th poss ssion of knowl dg carri s an thical r sponsibility.” E aluat this claim. 4. The traditional TOK diagram indicates four ways of knowing. Propose the inclusion of a fifth way of knowing selected from intuition, memory or imagination, and explore the knowledge issues it may raise in two areas of knowledge. 5. “That which can b ass rt d without idence can be dismiss d without id nc .” (Christoph r itch ns). Do you agree? 6. Can we know when to trust our emotions in the pursuit of knowledge? Consider history and one other area of knowledge. IB TOK ESSAY WRITERS HELP SOLUTION. 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