Mathematical Methods 2019 v1.2

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1 Problem-solving and modelling task (20%) This sample has been compiled by the QCAA to assist and support teachers to match evidence in student responses to the characteristics described in the instrument-specific marking guide (ISMG). Assessment objectives This assessment instrument is used to determine student achievement in the following objectives: 1. select, recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 2 and/or 3 2. comprehend mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or 3 3. communicate using mathematical, statistical and everyday language and conventions 4. evaluate the reasonableness of solutions 5. justify procedures and decisions by explaining mathematical reasoning 6. solve problems by applying mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or

2 Instrument-specific marking guide (ISMG) Criterion: Formulate Assessment objectives 1. select, recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 2 and/or 3 2. comprehend mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or 3 5. justify procedures and decisions by explaining mathematical reasoning The student work has the following characteristics: documentation of appropriate assumptions accurate documentation of relevant observations accurate translation of all aspects of the problem by identifying mathematical concepts and techniques. statement of some assumptions statement of some observations translation of simple aspects of the problem by identifying mathematical concepts and techniques. Marks does not satisfy any of the descriptors above. 0 Criterion: Solve Assessment objectives 1. select, recall and use facts, rules, definitions and procedures drawn from Unit 3 Topics 2 and/or 3 6. solve problems by applying mathematical concepts and techniques drawn from Unit 3 Topics 2 and/or 3 The student work has the following characteristics: accurate use of complex procedures to reach a valid solution discerning application of mathematical concepts and techniques relevant to the task accurate and appropriate use of technology. use of complex procedures to reach a reasonable solution application of mathematical concepts and techniques relevant to the task use of technology. use of simple procedures to make some progress towards a solution simplistic application of mathematical concepts and techniques relevant to the task superficial use of technology. Marks inappropriate use of technology or procedures. 1 does not satisfy any of the descriptors above. 0 Page 2 of 15

3 Criterion: Evaluate and verify Assessment objectives 4. evaluate the reasonableness of solutions 5. justify procedures and decisions by explaining mathematical reasoning The student work has the following characteristics: evaluation of the reasonableness of solutions by considering the results, assumptions and observations documentation of relevant strengths and limitations of the solution and/or model. justification of decisions made using mathematical reasoning. statements about the reasonableness of solutions by considering the context of the task statements about relevant strengths and limitations of the solution and/or model statements about decisions made relevant to the context of the task. Marks statement about a decision and/or the reasonableness of a solution. 1 does not satisfy any of the descriptors above. 0 Criterion: Communicate Assessment objective 3. communicate using mathematical, statistical and everyday language and conventions The student work has the following characteristics: correct use of appropriate technical vocabulary, procedural vocabulary, and conventions to develop the response coherent and concise organisation of the response, appropriate to the genre, including a suitable introduction, body and conclusion, which can be read independently of the task sheet. use of some appropriate language and conventions to develop the response adequate organisation of the response. Marks does not satisfy any of the descriptors above. 0 Page 3 of 15

4 Task Context Formulas can be used to model the position, velocity and acceleration of runners at any time during a race. The three models proposed for Competitors 1, 2 and 3 are: Competitor 1: dd = aa sin(bbbb) Competitor 2: vv = cc(1 ee ffff) + gg(1 ee htt ) Competitor 3: dd = 1+ee jjjj ( kkkk+ll) + mm where dd is the distance in metres, tt represents time in seconds and vv is the velocity in metres per second. aa, bb, cc, ff, gg, h, jj, kk, ll and mm are parameter values. The table below shows the 10-metre split times for the 100-metre race for Competitor 4. Position dd (metres) Elapsed time tt (seconds) The results for the race are: Competitor 1 comes fourth Competitor 2 comes second Competitor 3 comes third Competitor 4 wins the race. Page 4 of 15

5 Task Write a report that discusses the appropriateness of using mathematical functions to model the running of a 100-metre race. You will: use given function types to model the running of a 100-metre race by Competitors 1, 2 and 3 use data for Competitor 4 s 10-metre splits in the 100-metre race to develop a function that models the distance that the competitor has run at any time during the race provide a mathematical analysis of the race that includes when and where the competitors were running the fastest and slowest, and when and where the competitors were accelerating the most and the least. The following stages of the problem-solving and mathematical modelling approach should inform the development of your response. Once you understand what the problem is asking, design a plan to solve the problem. Translate the problem into a mathematically purposeful representation by first determining the applicable mathematical and/or statistical principles, concepts, techniques and technology that are required to make progress with the problem. Identify and document appropriate assumptions, variables and observations, based on the logic of a proposed model; include a description of how the parameters for the given race functions and the model for the data will be determined. In mathematical modelling, formulating a model involves the process of mathematisation moving from the real world to the mathematical world. Select and apply mathematical and/or statistical procedures, concepts and techniques previously learnt to solve the mathematical problem to be addressed through your model. Synthesise and refine existing models, and generate and test hypotheses with secondary data and information, as well as using standard mathematical techniques. Models should satisfy the rules for the final position in the race. Solutions can be found using algebraic, graphic, arithmetic and/or numeric methods, with and/or without technology. Once you have achieved a possible solution, consider the reasonableness of the solution and/or the utility of the model in terms of the problem. Evaluate your results and make a judgment about the solution/s to the problem in relation to the original issue, statement or question. This involves exploring the strengths and limitations of your model. Where necessary, this will require you to go back through the process to further refine the model/s. Check that the output of your model provides a valid solution to the real-world problem it has been designed to address. The model should appropriately represent the running of a race. This stage emphasises the importance of methodological rigour and the fact that problem-solving and mathematical modelling is not usually linear and involves an iterative process. The development of solutions and models to abstract and real-world problems must be capable of being evaluated and used by others and so need to be communicated clearly and fully. Communicate your findings systematically and concisely using mathematical, statistical and everyday language. Draw conclusions, discussing the key results and the strengths and limitations of the model/s. You could offer further explanation, justification and/or recommendations, framed in the context of the initial problem. Page 5 of 15

6 Sample response Criterion Allocated marks Marks awarded Formulate Assessment objectives 1, 2, 5 Solve Assessment objectives 1, 6 Evaluate and verify Assessment objectives 4, 5 Communicate Assessment objective Total The annotations show the match to the instrument-specific marking guide performance level descriptors. coherent and concise organisation of the response The introduction describes what the task is about and briefly outlines how the writer intends to complete the task. Introduction The purpose of this problem-solving and modelling task is to determine whether formulas can be used to model the position, velocity and acceleration of competitors at any time in a 100-metre sprint race. A set of sprint data and three suggested function types with unknown parameters have been provided. Each competitor must finish in a designated position, which must be supported by each model. Various forms of graphing technology will be used to determine appropriate models and solve equations that cannot be solved analytically. The models will be compared for accuracy and plausibility. The given equations are: Equations Competitor 1: dd = aaaaaaaa(bbbb) Competitor 2: vv = cc(1 ee ffff) + gg(1 ee htt ) Competitor 3: dd = jjjj (1+ee kkkk+ll ) + mm Competitor 4: equation modelled from data Place 4th 2nd 3rd 1st where dd is the position in metres, tt is the time in seconds and vv is the velocity in metres per second. Formulate [3 4] accurate documentation of relevant observations Formulate [3 4] documentation of appropriate assumptions Observations and assumptions The primary observation for this task was that the competitors ran a standard 100-metre sprint. It was assumed all runners intended to win. This frames the models with a realistic outcome, which is useful for real-life applications. With this observation, the following assumptions are deduced: 1. Running conditions are perfect. This assumption incorporates factors such as fine weather, no head wind, etc. 2. Competitors 1, 2 and 3 are Olympic podium finishers. According to the data (see Competitor 4 data, page 9), Competitor 4 finished the 100-metre sprint in 9.58 seconds. Therefore, the assumption is made that Competitors 1, 2 and 3 also Page 6 of 15

7 finish the race in approximately 10 seconds. This is a valid assumption because, since the 1996 Olympics, all podium places for the 100-metre sprint finished the race in less than 10 seconds (Olympic Games 2017). 3. Competitors run a standard race with no false start. A competitor false starting and not getting stopped for it would cause the position to be greater than 0 metres at 0 seconds. This would affect the models and the appropriateness of comparing competitors. Theoretically, a sprint athlete could sprint at 40 miles/hour (which equates to metres/second) (Live Science 2010). Therefore, it is assumed that no sprinter can exceed metres/second. 4. Competitors use blocks to begin the race and accelerate from them. coherent and concise organisation of the response, appropriate to the genre Formulate [3 4] accurate translation of all aspects of the problem by identifying mathematical concepts and techniques Mathematical concepts and procedures The online graphing program Desmos will be used to determine the unknown parameters because this is an efficient way to visually see how the parameters transform an equation. Calculus procedures will also be used to determine the velocity and acceleration functions. The maximum and minimum velocity and acceleration will be calculated by: Max./min. velocity solve for tt when the acceleration equals 0 Max./min. acceleration solve for tt when the derivative of acceleration equals 0 and then consideration will be given to whether this value represents a global or local optimal value. In some cases, analytical procedures will be used to calculate values and, when this is not possible, technology will be used. The domain for all functions used time (tt) values greater than or equal to zero and less than 11 seconds (0 tt 11). Determining the models Competitor 1: dd 11 (tt) = aaaaaaaa (bbbb) accurate and appropriate use of technology Using the graphing software, parameter values were changed and refined to arrive at a feasible model (found when aa = 157 and bb = 0.069). The displacement time graph and table for this model are: correct use of appropriate technical vocabulary, procedural vocabulary, and conventions to develop the response Distance (m) Graph of data values Competitor Time (s) Page 7 of 15

8 To determine the exact time Competitor 1 crosses the finish, the function was equated to 100 and the equation solved. dd 1 (tt) = 157 sin(0.069tt) 100 = 157sin (0.069tt) 0.64 = sin(0.069tt) ssssss 1 (0.64) = 0.069tt ssssss 1 (0.64) = tt tt = seconds Competitor 1 placed 4th in the race, so this represents the slowest time. Using calculus procedures, the displacement function was differentiated to find velocity and the second derivative modelled acceleration. correct use of appropriate technical vocabulary, procedural vocabulary, and conventions coherent and concise organisation of the response vv 1 (tt) = cos (0.069tt) and aa 1 (tt) = 0.747sin (0.069tt) where vv 1 (tt) and aa 1 (tt) denote velocity and acceleration for Competitor 1. The table of values shows the velocity and acceleration at time tt. Time tt (seconds) Velocity (m/s) Acceleration (m/s 2 ) accurate and appropriate use of technology The times for the maximum and minimum velocity and acceleration are: Velocity Acceleration tt (sec) vv (m/s) tt (sec) aa (m/s 2 ) Maximum Minimum These values were found by graphing the functions and analysing the graph to determine the global maximum and minimum. Page 8 of 15

9 Competitor 2: vv = cc(11 ee ffff) + gg(11 ee hhhh ) The model given for Competitor 2 is a velocity function; therefore, the integral must be found to model displacement. dd 2 (tt) = vv dddd dd 2 (tt) = cc(1 ee ffff) + gg(1 ee htt ) dddd dd 2 (tt) = cc ccee ffff + gg ggee htt dddd Solve [4 5] accurate and appropriate use of technology dd 2 (tt) = cccc ccee ffff ff + gggg ggeehtt h + nn dd 2 (tt) = cccc + cccc ffff gggghtt + gggg + nn where nn is the constant of integration. ff h Using Desmos produced the parameter values cc = 10.5, ff = 10, gg = 0.105, h = 0.1 and nn = 0 The displacement time graph and table of values for this function are below: Distance (m) 120 Competitor Time (s) discerning application of mathematical concepts and techniques relevant to the task Writer has recognised the appropriate technique to use to solve. The intersection with dd = 100 (blue line) and the displacement function (red line) shows that Competitor 2 will finish the race in 9.69 seconds and is placed 2nd. The velocity and acceleration models are vv 2 (tt) = 10.5(1 ee 10tt ) (1 ee 0.1tt ) and aa 2 (tt) = 105ee 10tt ee 0.1tt At time zero, the velocity is zero, which would be the case in a sprinting race. Page 9 of 15

10 The table of values for both velocity and acceleration is: vv 2 (tt) aa 2 (tt) The times for the maximum and minimum velocity and acceleration are: Velocity Acceleration tt (sec) vv (m/s) tt (sec) aa (m/s 2 ) Maximum Minimum Note that at time zero, the sprinter has the greatest acceleration when leaving the blocks. Competitor 3: dd = jjjj (1+ee kkkk+ll ) + mm The model for Competitor 3 is a logistical equation. Using Desmos, the following 10.3tt function was generated: dd 3 (tt) = 1+ee ( 0.47tt+1) The displacement at certain times and graph of the function are given below: Distance (m) Competitor Time (s) Page 10 of 15

11 Using graphical methods, it was found that Competitor 3 finishes the race in 9.95 seconds and is placed 3rd. The derivative of displacement is required to determine the velocity. The quotient rule was used to determine the velocity function. vv 3 (tt) = ee( 0.47tt+1) 10.3t( 0.47ee (.47tt+1) ) (1 + ee ( 0.47tt+1) ) 2 Simplifying by expanding and factorising: vv 3 (tt) = ee( 0.47tt+1) ttee (.47tt+1) = ee ( 0.47tt+1) ttee (.47tt+1) (1+ee ( 0.47tt+1) ) ttee (.47tt+1) (1+ee ( 0.47tt+1) ) 2 To determine the function for acceleration: (1+ee ( 0.47tt+1) ) 2 = ee ( 0.47tt+1) + Let ww = (4.841ttee( 0.47tt+1) ) and zz = 10.3 and differentiate separately (1+ee ( 0.47tt+1) ) 2 (1+ee ( 0.47tt+1) ) ww = (4.841ttee( 0.47tt+1) ) (1+ee ( 0.47tt+1) ) 2 using product rule ww = ppqq + qqqq pp = (4.841ttee ( 0.47tt+1) ) pp = 2.275ttee ( 0.47tt+1) ee ( 0.47tt+1) qq = 1 + ee ( 0.47tt+1) 2 qq = 0.94ee ( 0.47tt+1) (1 + ee ( 0.47tt+1) ) 3 correct use of appropriate technical vocabulary, procedural vocabulary, and conventions to develop the response Calculus notation and equality signs used appropriately. ww = 4.841ttee ( 0.47tt+1) 0.94ee ( 0.47tt+1) 1 + ee ( 0.47tt+1) ee ( 0.47tt+1) ttee ( 0.47tt+1) ee ( 0.47tt+1) ww = 4.56ttee2( 0.47tt+1) (1 + ee ( 0.47tt+1) ) ttee( 0.47tt+1) ee ( 0.47tt+1) (1 + ee ( 0.47tt+1) ) 2 zz = 10.3 (1+ee ( 0.47tt+1) ) using product rule zz = rrss + ssss rr = 10.3 rr = 0 ss = (1 + ee ( 0.47tt+1) ) 1 ss = 0.47ee ( 0.47tt+1) (1 + ee ( 0.47tt+1) ) 2 zz = 4.841ee( 0.47tt+1) (1 + ee ( 0.47tt+1) ) 2 accurate use of complex procedures to reach a valid solution Differentiation and factorisation procedure evident. The solution involves a combination of parts that are interconnected Combine parts ww and zz aa 3 (tt) = 4.56ttee2( 0.47tt+1) (1 + ee ( 0.47tt+1) ) ttee( 0.47tt+1) ee ( 0.47tt+1) (1 + ee ( 0.47tt+1) ) ee( 0.47tt+1) (1 + ee ( 0.47tt+1) ) 2 aa 3 (tt) = 4.56ttee2( 0.47tt+1) (1 + ee ( 0.47tt+1) ) ttee( 0.47tt+1) ee ( 0.47tt+1) 3 (1 + ee ( 0.47tt+1) ) 2 ee ( 0.47tt+1) 4.56ttee( 0.47tt+1) aa 3 (tt) = (1 + ee ( 0.47tt+1) ) 2 ( 0.47tt+1) 2.275tt ee Page 11 of 15

12 The table of values for both functions is: vv 3 (tt) aa 3 (tt) coherent and concise organisation of the response which can be read independently of the task sheet Formulate [3 4] accurate documentation of relevant observations Formulate [3 4] documentation of appropriate assumptions accurate and appropriate use of technology discerning application of mathematical concepts Writer has forced the initial values for position to zero and minimised error for finishing time. Using these models: Velocity Acceleration tt (sec) vv (m/s) tt (sec) aa (m/s 2 ) Maximum Minimum Competitor 4 Given data was used to produce a model for Competitor 4 (data on next page). When the data was plotted, it steadily increased from left to right and a non-linear polynomial was assumed. Using a spreadsheet program produced the following regression functions (forced through the point (0,0)). As a general rule, the higher the coefficient of determination, RR 2 the more useful the model. Polynomial Regression model (dd) RR 22 Linear tt Quadratic tt tt Cubic.0747tt tt tt Fourth degree. 0092tt tt tt tt 1 Fifth degree.0013tt tt tt tt tt Sixth degree. 0003tt tt tt tt tt tt 1 Evaluate and verify [4 5] evaluation of the reasonableness of solutions The RR 2 value is very misleading as the fourth-, fifth- and sixth-degree polynomial models were deemed a perfect fit. Comparing the actual data values to the values generated using the model showed discrepancies. Competitor 4 was the winner of the race with a time of 9.58 seconds. The fourth-degree polynomial was chosen as the model for Competitor 4 as the time to complete the race was 9.59 seconds, which compared well to the actual time of 9.58 seconds. The residual error analysis below ( ss aaaaaaaaaaaa ss uuuuuuuugg mmmmmmmmmm 100%), produced in a spreadsheet program, shows ss aaaaaaaaaaaa nearly all values are less than 1% off. Page 12 of 15

13 accurate and appropriate use of technology Appropriate formulas were used to determine ss uuuuuuuuuu mmmmmmmmmm, e.g. given that time values are in column A, distance values in column B, using fourth-degree polynomial values are in column C, percentage error values are in column D and time 1.89 is in position A3, the value was generated using the equation = *AA *AA *AA *AA3. Similarly, the percentage error value was generated using the equation BB3 DD BB3 Given: ss 4 (tt) =.0092tt tt tt tt, using calculus methods: accurate use of complex procedures to reach a valid solution The solution consists of an involved combination of parts that are interconnected vv 4 (tt) =.0368tt tt tt aa 4 (tt) =.1104tt tt To find the local maximum and minimum velocities: vv 4(tt) = 0. 0 =.1104tt tt Using the quadratic formula: tt = ± ((.15042) ) tt 5.59, 8.03 To determine the nature of the optimal values, the gradient to the left and the right of these values was found: vv 4 (5) =.199 and vv 4 (7).1598 and vv 4 (. 805) =.1507 The gradient is positive to the left and negative to the right of tt = 5.59; therefore, a local maximum velocity occurs at time tt = 5.59 seconds and, using similar reasoning, a local minimum at time tt = 8.03 seconds. discerning application of mathematical concepts However, it is clear from the graph of the velocity model that the maximum velocity occurs at the end of the race at vv 4 (9.59) = metres/second, and the minimum value occurs at the start vv 4 (0) = metres/second. Page 13 of 15

14 accurate and appropriate use of technology Using similar reasoning, the maximum acceleration occurs at the start of the race and the minimum during the race. The table below summarises the findings: Velocity Acceleration tt (sec) vv (m/s) tt (sec) aa (m/s 2 ) Maximum Minimum Evaluation A major factor underpinning this task is how realistic the models are and if they can be used reliably to interpolate values that are plausible. This will allow the validity of the model to be tested. Evaluate and verify [4 5] documentation of relevant strengths and limitations of the solution and/or model Evaluate and verify [4 5] documentation of relevant strengths and limitations of the solution and/or model Displacement with respect to time All models satisfied the mathematical assumptions stated; that is, the initial positions were 0 metres and all four competitors finished the race in approximately 10 seconds. All models could be used to determine a position of a competitor at any time during the race and yield a plausible value. Velocity with respect to time Competitor 1, Competitor 3 and Competitor 4 all have initial velocities greater than 0 metres/second (and a standing start was assumed). Competitor 2 s proposed model was a velocity function; therefore, parameters were determined to ensure that the initial velocity would equal zero. The second limiting factor is the plausibility of the velocities determined from the model during the race. The Competitor 1 model is completely unrealistic, as the sprinter continues to slow down from the start. The Competitor 2 model would produce a sharp increase in velocity in a short amount of time, i.e. after a standing start, the sprinter is moving at m/s after one second, and then continues to run at reduced velocities for the remainder of the race until the finish, when they are running at a rate of m/s. Competitor 4 steadily increases their velocity until time 5.59 seconds, slows, and then increases their velocity until the end of the race. This at least resembles how a runner may run a race. Page 14 of 15

15 Evaluate and verify [4 5] documentation of relevant strengths and limitations of the solution and/or model Acceleration with respect to time The acceleration values for Competitor 1 are constantly decreasing from an initial value of zero. The competitor never accelerates. The use of a trigonometric function to model a race is invalid. Competitor 2 has an enormous initial acceleration that is not realistic. Competitor 3 accelerates and decelerates throughout the race. The Competitor 4 model has a plausible initial acceleration and a slow deceleration throughout the race until near the finish, when they accelerate again. coherent and concise organisation of the response The conclusion summarises the report, giving information about the problem that had to be solved, the mathematical processes used to solve the problem and discussion about the results, including any problems encountered and conclusions drawn from the information presented in tables and graphs. Conclusion Using any function to model the instantaneous position, velocity and acceleration of a runner at any point in a 100-metre race is problematic. Models were obtained by fitting functions to data (as with Competitor 4), and using given functions (Competitor 1, 2 and 3). The validity of all models was tested using the correlation coefficient, residual analysis and the real-world application. All models had limitations. The polynomial model from the given data has the potential to produce a plausible model. A suggestion would also be to produce a model for displacement, and then a separate model for average velocity, as opposed to instantaneous velocity. In this way, more realistic values for these variables at different times in the race could be modelled. Reference list Live Science 2010, Humans Could Run 40 mph, in Theory, Olympic Games 2017, 100m Men, Page 15 of 15