# Ib Math Ia

MATME/PF/M12/N12/M13/N13 MATHEMATICS Standard Level The portfolio – tasks For use in 2012 and 2013 © International Baccalaureate Organization 2010 7 pages For final assessment in 2012 and 2013 2 MATME/PF/M12/N12/M13/N13 C O N T E N TS T y p e I t as k s Circles T y p e I I t as k s Fish Production Gold Medal Heights INTRODUC TI ON W h a t is t h e p u r p ose of t h is d oc u m e n t ? This document contains new tasks for the portfolio in mathematics SL.These tasks have been produced by the IB, for teachers to use in the examination sessions in 2012 and 2013.

It should be noted that most tasks previously produced and published by the IB will no longer be valid for assessment after the November 2010 examination session.These include all the tasks in any teacher support material (TSM), and the tasks in the document tfolio tasks 2009 The tasks in the in the 2012 examinations but N O T in 2013.

**Ib Math Ia**

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Copies of all TSM tasks published by the IB are available on the Online Curriculum Centre (OCC), under Internal Assessme not be used, even in slightly modified form. W h a t h a p p e n s i f t e a c h e r s u s e t h e s e o l d t a s k s?

The inclusion of these old tasks in the portfolio will make the portfolio non -compliant, and such portfolios will therefore attract a 10-mark penalty. Teachers may continue to use the old tasks as practice tasks, but they should not be included in the portfolio for final assessment. W h a t o t h e r d oc u m e n t s s h o u l d I u se? All teachers should have copies of the mathematics SL subject guide (second edition, September 2006), including the teaching notes appendix, and the TSM (September 2005). Further information, ncluding additional notes on applying the criteria, is available on the Online Curriculum Centre (OCC). Important news items are also available on the OCC, as are the diploma programme coordinator notes, which contain updated information on a variety of issues. W h i c h t as k s c a n I u se i n 2012? The only tasks produced by the IB that may be submitted for assessment in 2012 are the ones contained in this document, and those in the document Portfolio tasks 2011 2012 . There is no requirement to use tasks produced by the IB, and there is no date restriction on tasks written by teachers.

For final assessment in 2012 and 2013 3 MATME/PF/M12/N12/M13/N13 C a n I u se t h ese t as k s b e f o r e M a y 2012? These tasks should only be submitted for final assessment from May 2012 to November 2013. Students should not include them in portfolios before May 2012. If they are included, they will be subject to a 10-mark penalty. Please note that these dates refer to examination sessions, not when the work is completed. W h i c h t as k s c a n I u se i n 2013? The only tasks produced by the IB that may be submitted for assessment in 2013 are the ones contained in this document. T e c h n ology

There is a wide range of technological tools available to support mathematical work. These include graphic display calculators, Excel spreadsheets, Geogebra, Autograph, Geometer sketch pad and Wolframalpha. Many are free downloads from the Internet. Students (and teachers) should be encouraged to explore which ones best support the tasks that are assigned. Teachers are reminded that good technology use should enhance the development of the task. E x t r a c ts f r o m d i p lom a p r og r a m coo r d i n a to r n ot es Important information is included in the DPCN, available on the OCC.

Teachers should ensure they are familiar with these, and in particular with the ones noted below. Please note that the reference to the 2009/2010 document is outdated. C op i es of t as k s a n d m a r k i n g/sol u t ion k e ys Teachers are advised to write their own tasks to fit in with their own teaching plans, to select from the 2009/2010 document, or to use tasks written by other teachers. In each case, teachers should work the task themselves to make sure it is suitable, and provide a copy of the task, and an answer, solution or marking key for any task submitted.

This will help the moderators confirm the levels awarded by the teacher. It is particularly important if teachers modify an IB published task to include a copy of the modified task. While this is permitted, teachers should think carefully about making any changes, as the tasks have been written with all the criteria in mind, to allow students to achieve the higher levels. N on -co m p l i a n t po r t f ol ios f r om M a y 2012 Please note the following information on how to deal with portfolios that do not contain one task of each type. This will be applied in the May 2012 and subsequent examination sessions.

If two pieces of work are submitted, but they do not represent a Type I and a Type II task (for example, they are both Type I or both Type II tasks), mark both tasks, one against each Type. For example, if a candidate has submitted two Type I tasks, mark one using the Type I c riteria, and the other using the Type II Criteria. Do not apply any further penalty This means that the current system of marking both tasks against the same criteria and then applying a penalty of 10 marks will no longer be used. For final assessment in 2012 and 2013 4 MATME/PF/M12/N12/M13/N13 SL T YPE I

A im : In this t ask you wi l l conside r a se t of numbe rs tha t a r e pr esent ed in a symme t r i c a l pa t t e rn. 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 E n ( r ) be the ( r 1) th element in the nth row, starting with r 15 Example: E5 (2) . 9 0. Find the general statement for E n ( r ) .

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. For final assessment in 2012 and 2013 5 MATME/PF/M12/N12/M13/N13 C IR C L ES SL T YPE I A im : The a im of this t ask is to invest i ga t e posi t ions of points in i nt e rse c t ing c i r c l es. The following diagram shows a circle C 1 with centre O and radius r, and any point P. r P O C1 The circle C 2 has centre P and radius OP. Let A be one of the points of intersection of C 1 and C 2 . Circle C 3 has centre A, and radius r.

The point P is the intersection of C 3 with (OP). This is shown in the diagram below. C3 A O P’ P C2 C1 Let r 1 . Use an analytic approach to find OP , when OP 2 , OP 3 and OP 4 . Describe what you notice and write a general statement to represent this. Let OP 2 . Find OP , when r 2 , r 3 and r 4 . 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 general statement for OP . 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. For final assessment in 2012 and 2013 6 MATME/PF/M12/N12/M13/N13 F IS H PR O D U C T I O N SL T YPE II A im: This t ask conside rs comme r c i a l f ishing in a pa r t i cul a r count ry in two di ff e r ent envi ronments the se a and f ish f a rms (aqua cul tur e). The da t a is t a k en f rom the U N St a t ist i cs D ivisi on C ommon D a t a b a se . The following table gives the total mass of fish caught in the sea, in thousands of tonnes (1 tonne = 1000 kilograms). Y ea r

T ot a l M ass 1980 426. 8 1981 470. 2 1982 503. 4 1983 557. 3 1984 564. 7 1985 575. 4 1986 579. 8 1987 624. 7 1988 669. 9 Y ea r T ot a l M ass 1989 450. 5 1990 379. 0 1991 356. 9 1992 447. 5 1993 548. 8 1994 589. 8 1995 634. 0 1996 527. 8 1997 459. 1 Y ea r T ot a l M ass 1998 487. 2 1999 573. 8 2000 503. 3 2001 527. 7 2002 566. 7 2003 507. 8 2004 550. 5 2005 426. 5 2006 533. 0 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. Y ea r T ot a l M ass 1980 1. 4 1981 1. 5 1982 1. 7 1983 2. 0 1984 2. 2 1985 2. 7 1986 3. 1 1987 3. 3 1988 4. 1 Y ea r T ot a l M ass 1989 4. 4 1990 5. 8 1991 7. 8 1992 9. 1 1993 12. 4 1994 16. 0 1995 21. 6 1996 33. 2 1997 5. 5 Y ea r T ot a l M ass 1998 56. 7 1999 63. 0 2000 79. 0 2001 67. 2 2002 61. 2 2003 79. 9 2004 94. 7 2005 119. 8 2006 129. 0 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. For final assessment in 2012 and 2013 7

MATME/PF/M12/N12/M13/N13 G O L D M E D A L H E I G H TS SL T YPE II A i m : T he a i m of th i s t a sk i s to O lympi c G ames. high jump in the The table below gives the height (in centimeters) achieved by the gold medalists at various Olympic Games. 1932 Y ea r H e igh t (c m) 197 1936 203 1948 198 1952 204 1956 212 1960 216 1964 218 1968 224 1972 223 1976 225 1980 236 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. 1896 1904 1908 1912 1920 1928 1984 1988 1992 1996 2000 2004 2008 Y ea r H eigh t (c m) 190 180 191 193 193 194 235 238 234 239 235 236 236 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? For final assessment in 2012 and 2013