INTERNATIONAL BACCALAUREATE ORGANIZATION Diploma Programme GROUP 5 MATHEMATICS Solutions Using a Graphic Display Calculator
Group 5 Mathematics Solutions Using a Graphic Display Calculator August 2001 h International Baccalaureate Organization 2001 International Baccalaureate Organization Route des Morillons 15 1218 Grand-Saconnex Geneva, SWITZERLAND
Contents Introduction 2 Advice to Candidates 2 Markscheme Instructions 3 Minimum Requirements for Examinations in Mathematics HL, Mathematical Methods SL and Further Mathematics SL 4 Example 1 Mathematical methods SL specimen paper 1, question 6 5 Example 2 Mathematical HL specimen paper 1, question 11 6 Example 3 Mathematics HL specimen paper 1, question 19 7 Example 4 Mathematical methods SL specimen paper 2, question 1(b) 8 Example 5 Mathematical methods SL, paper 2, November 2000, extract from question 4 9
Introduction This document has been produced in response to queries from teachers about the use of graphic display calculators in examinations. Teachers have asked what working should be shown by candidates to indicate their use of calculators in answering examination questions. This query is almost impossible to answer in a general way, as it would depend on the particular question, but the examples in this document demonstrate the type of working candidates should be encouraged to show when using graphic display calculators. The questions are taken from the specimen papers produced in 1999 and one examination paper from November 2000. There is a mixture of paper 1 and paper 2 questions from mathematics HL and mathematical methods SL papers. After each example, there are notes intended for examiners which will help illustrate how part marks are awarded for incorrect answers. Advice to Candidates Teachers should advise candidates: to explain what they are doing, in mathematical terms, wherever possible to avoid using calculator commands, eg fnint (one of the reasons being that these commands may be meaningless to an examiner who has a different brand of calculator) to show any mathematical manipulation which is required before inputting data into the calculator, eg in statistics, if standardized data is required before calculating probabilities, candidates should write down the standardized values. Candidates should be encouraged to show their working, especially in paper 2. This will enable examiners to reward understanding, even if the answer is incorrect. Candidates who simply write down an incorrect answer, with no indication of how they obtained it, lose marks. To try to avoid this happening, examiners attempt to write questions that require intermediate answers. To order the following items, contact the sales department at IBCA via the IBO s web site, www.ibo.org, or by e-mail on sales@ibo.org. Copies of individual examination papers and markschemes for any of the group 5 courses. Packs of examination papers and packs of markschemes for an examination session. Booklets of specimen papers and markschemes for each subject (published in 1999). 2 Solutions Using a Graphic Display Calculator, August 2001
Markscheme Instructions The markschemes contain instructions to examiners about rewarding the use of a graphic display calculator. On paper 1, full marks are awarded for the correct answer, irrespective of the method used. On paper 2, examiners are told when it is acceptable to write down answers from a calculator without showing working. The following is an extract from the markscheme instructions. It shows the abbreviations used in the notes to each example in this document. Abbreviations The markscheme may make use of the following abbreviations: M Marks awarded for Method A Marks awarded for an Answer or for Accuracy G Follow Through (ft) Marks Marks awarded for correct solutions, generally obtained from Graphic Display Calculator, irrespective of working shown Errors made at any step of a solution can affect all working that follows. To limit the severity of the penalty, follow through (ft) marks should be awarded. The procedures for awarding these marks require that all examiners: (i) penalize the error when it first occurs; (ii) (iii) Alternative Solutions accept the incorrect answer as the appropriate value or quantity to be used in all subsequent working; award M marks for a correct method and A (ft) marks if the subsequent working contains no further errors. Alternative solutions are indicated by OR. Where these are accompanied by G marks, they usually signify that the answer is acceptable from a graphic display calculator without showing working. For example: Mean = 7906/134 (M1) = 59 (A1) OR Mean = 59 Graphic Display Calculators (G2) Many candidates will be obtaining solutions directly from their calculators, often without showing any working. They have been advised that they must use mathematical notation, not calculator commands when explaining what they are doing. Incorrect answers without working will receive no marks. However, if there is written evidence of using a graphic display calculator correctly, method marks may be awarded. Where possible, examples will be provided to guide examiners in awarding these method marks. Solutions Using a Graphic Display Calculator, August 2001 3
Minimum Requirements for Examinations in Mathematics HL, Mathematical Methods SL and Further Mathematics SL There are many different calculators being used for IBO examinations. Each calculator has strengths and weaknesses. Teachers are advised to be aware of the capabilities of the calculators used by all their students. All calculators should meet the minimum requirements specified below. While every effort is made to update the list of examples of calculators which meet these requirements (and those calculators not allowed in IBO examinations), it is not possible to produce a definitive list. Teachers who have queries about a particular calculator should contact the subject area manager for mathematics at IBCA. From May 2001, candidates are expected to have access to graphic display calculators which can: draw graphs with any viewing window solve equations using the inbuilt equation solver functions of the calculator add and multiply matrices, and find inverse matrices determine a numerical derivative at a point calculate a numerical integral between given limits. Examination questions will be written assuming that candidates have access to graphic display calculators with these functions. In addition, for the statistics options, teachers are advised to encourage the use of a calculator with appropriate built-in statistical functions. 4 Solutions Using a Graphic Display Calculator, August 2001
EXAMPLE 1 A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is 0.4t modelled by N = 10e. (a) How many leopards are there after 2 years? (b) How long will it take for the number of leopards to reach 100? Give your answers to an appropriate degree of accuracy. Notes: (a) (b) Award M1 for the diagram or for N = 22.3, and A1 for the answer 22. Award M1 for indicating that the solution may be found by solving 0.4t N = 10e and N = 100. Solutions Using a Graphic Display Calculator, August 2001 5
EXAMPLE 2 Solve, by any method, the following system of equations: 3x 2y + z = 4 x + y z = 2 2x + 3y = 4 Notes: Award M1 for the matrix equation and M1 for x =A -1 b. 6 Solutions Using a Graphic Display Calculator, August 2001
EXAMPLE 3 2 Find the area of the region enclosed by the graphs of y = sin x and y = x 2x + 1.5, where 0 x π. Notes: Award M1 for subtracting the equations, and A1 for one correct limit (0.662, 1.70). Solutions Using a Graphic Display Calculator, August 2001 7
EXAMPLE 4 The following diagram represents the lengths, in cm, of 80 plants grown in a laboratory. frequency 25 20 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 110 length (cm) Calculate estimates for the mean and the standard deviation of the lengths of the plants. Notes: Award M1 for each correct line in the distribution table. Award M0 M1 for using values other than the mid-interval values for x and the correct frequencies and A1 (ft) for each correct follow through answer. 8 Solutions Using a Graphic Display Calculator, August 2001
EXAMPLE 5 The Cartesian equations of the paths of two aircraft are 5x + 12y = 224 and 12x 5y = 301 respectively. Find the coordinates of the point where the two paths cross. Notes: Award M1 for each correct re-arranged equation, and A1 (ft) for each correct follow through answer. Solutions Using a Graphic Display Calculator, August 2001 9