Mathematical Methods 2020 v1...200041 Mathematical Methods 2020 v1.2 IA1 mid-level annotated sample response February 2020 Problem-solving and modelling task (20%) This sample has

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Mathematical Methods 2020 v1.2 IA1 mid-level annotated sample response February 2020

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 3.

Mathematical Methods 2020 v1.2 IA1 mid-level annotated sample response

Queensland Curriculum & Assessment Authority February 2020

Page 2 of 14

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: Marks

• documentation of appropriate assumptions • accurate documentation of relevant observations • accurate translation of all aspects of the problem by identifying mathematical concepts and

techniques.

3–4

• statement of some assumptions • statement of some observations • translation of simple aspects of the problem by identifying mathematical concepts and

techniques.

1–2

• 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


• 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.

6–7

• use of complex procedures to reach a reasonable solution • application of mathematical concepts and techniques relevant to the task • use of technology.

4–5

• 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.

2–3

• inappropriate use of technology or procedures. 1




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Criterion: Evaluate and verify

Assessment objectives 4. evaluate the reasonableness of solutions

5. justify procedures and decisions by explaining mathematical reasoning


• 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.

4–5

• 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.

2–3

• statement about a decision and/or the reasonableness of a solution. 1


Criterion: Communicate

Assessment objective 3. communicate using mathematical, statistical and everyday language and conventions


• 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.

3–4

• use of some appropriate language and conventions to develop the response • adequate organisation of the response.

1–2




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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− 𝑒𝑒ℎ𝑓𝑓) • Competitor 3: 𝑑𝑑 = 𝑗𝑗𝑓𝑓

1+𝑒𝑒(−𝑘𝑘𝑘𝑘+𝑙𝑙) +𝑚𝑚 where 𝑑𝑑 is the distance in metres, 𝑏𝑏 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 𝑏𝑏 (seconds) 0 0 10 1.89 20 2.88 30 3.78 40 4.64 50 5.47 60 6.29 70 7.10 80 7.92 90 8.75 100 9.58

The results for the race are: • Competitor 1 comes fourth • Competitor 2 comes second • Competitor 3 comes third • Competitor 4 wins the race.



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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.



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Sample response Criterion Allocated marks Marks awarded

Formulate Assessment objectives 1, 2, 5 4 3

Solve Assessment objectives 1, 6 7 4

Evaluate and verify Assessment objectives 4, 5 5 2

Communicate Assessment objective 3 4 4

Total 20 13

The annotations show the match to the instrument-specific marking guide performance level descriptors.

Communicate [3–4] 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. Formulate [3–4] accurate translation of all aspects of the problem … Communicate [3–4] correct use of appropriate technical vocabulary … Calculus and subscript notation, inequality signs and technical vocabulary are used appropriately.

Solve [4–5] application of mathematical concepts relevant to the task Shows understanding of calculus concepts and principles relevant to an aspect of the problem.

Introduction The aim of this report is to determine the appropriateness of using particular functions to model the running of a 100-metre race. Opportunity is provided to use given function types and to develop an appropriate model using given data. Using technology, parameter values for the given function types will be determined and a model for some given data will be found. Analytic procedures such as rules of differentiation will also be used to analyse aspects of the problem. The models must satisfy final places in the race assigned to each competitor; that is, Competitor 4 wins the race, Competitor 2 comes second, Competitor 3 comes third and Competitor 1 comes last. Given function types were used for Competitors 1, 2 and 3. The analysis begins with Competitor 2.

Displacement, velocity and acceleration model — Competitor 2 The given model for Competitor 2 defines the velocity of Competitor 2 at any time. The model is 𝑣𝑣 = 𝑐𝑐(1− 𝑒𝑒−𝑓𝑓𝑓𝑓) + 𝑔𝑔(1− 𝑒𝑒ℎ𝑓𝑓), where 𝑣𝑣 is in metres/second and time 𝑏𝑏 is in seconds. To generate a displacement model for Competitor 2, the integral of the velocity function must be determined.

𝑑𝑑Competitor 2 = �𝑐𝑐 − 𝑐𝑐𝑒𝑒−𝑓𝑓𝑓𝑓 + 𝑔𝑔 − 𝑔𝑔𝑒𝑒ℎ𝑓𝑓 𝑑𝑑𝑏𝑏

𝑑𝑑Competitor 2 = (𝑐𝑐 + 𝑔𝑔)𝑏𝑏 + 𝑐𝑐𝑒𝑒−𝑓𝑓𝑘𝑘

𝑓𝑓− 𝑔𝑔𝑒𝑒ℎ𝑘𝑘

ℎ+ 𝑛𝑛 where 𝑛𝑛 represents the constant for the

indefinite integral, 𝑑𝑑 is in metres and time 𝑏𝑏 is in seconds.



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Solve [6–7] accurate and appropriate use of technology Use of technology to change parameter values, generate an initial model and produce a sketch of the function.

Formulate [3–4] documentation of appropriate assumptions Further assumption has been stated.

Evaluate and verify [2–3] statement about the reasonableness of solutions by considering the context … Writer has considered the reasonableness of the solution to Competitor 2 model.

Competitor 2 places second in the race; therefore, values for the parameters 𝑐𝑐, 𝑓𝑓,𝑔𝑔, ℎ and 𝑛𝑛 must be determined if 𝑏𝑏 > 9.58 when 𝑑𝑑 = 100 (given race statistics for Competitor 4 indicate the winning time). The Desmos graphic calculator and sliders were used to determine appropriate (reasonable) values for the parameters (assuming the competitor does not run backwards at any time). The sketch below represents a first attempt to find suitable values.

The parameter values used for this sketch are: 𝑐𝑐 = −0.4,𝑔𝑔 = 1.6,𝑓𝑓 = −0.4,ℎ = −1.1 and 𝑛𝑛 = −1.7 A further assumption was identified — that all competitors are at the starting line at the start of the race as there is no handicapping for the event. To this end, the point (0, 0.755) was not realistic and a new model was developed. 𝑑𝑑Competitor 2 = 1.2𝑏𝑏 + 𝑒𝑒 .4𝑓𝑓 + 1.45455𝑒𝑒−1.1𝑓𝑓 − 2.455 Substituting the parameter values into the velocity model for Competitor 2: 𝑣𝑣Competitor 2 = −0.4(1− 𝑒𝑒0.4𝑓𝑓) + 1.6(1− 𝑒𝑒−1.1𝑓𝑓).

Time (secs)

Communicate Race statistics have not been shown and this affects the reader’s ability to read the response independently of the task sheet.



Page 8 of 14

Communicate [3–4] coherent and concise organisation of the response … Use of table to summarise findings.

Formulate [1–2] statement of some observations Formulate [3–4] documentation of appropriate assumptions Further assumptions have been stated.

Communicate [3–4] correct use of appropriate technical vocabulary, procedural vocabulary, and conventions … Appropriate convention used.

Graphical display has been labelled appropriately.

The values for this function are given in Table 1.

Table 1 Time 𝑏𝑏 (seconds) Velocity 𝑣𝑣 (metres/second) 0 0 1 1.26414 2 1.9129 3 2.46903 4 3.16156 5 4.149 6 5.607 7 7.77713 8 11.01277 9 15.8392 10 23.039233 11 33.7803

Researching statistics for the running of 100-metre races suggested that, at most, competitors have run at a rate of 12.2 metres/second and, therefore, my model was not valid for times at the end of the race (also, the model suggested that at the start of the race Competitor 2 was running very slowly). The analysis was repeated with new assumptions in mind; namely: • the velocity of the runner must not exceed 13 metres/second • the velocity should not be less than 5 metres/second at any point during the race. The parameter values were changed. The model and sketch for velocity of Competitor 2 are given below. 𝑣𝑣Competitor 2 = −3.9(1− 𝑒𝑒0.1𝑓𝑓) + 4.8(1− 𝑒𝑒−6.2𝑓𝑓) 𝑣𝑣Competitor 2 = 0.9 + 3.9𝑒𝑒0.1𝑓𝑓 − 4.8𝑒𝑒−6.2𝑓𝑓

Time (seconds)

met

res/

seco

nd



Page 9 of 14


The values for this function are given in Table 2.

Table 2 Time 𝑏𝑏 (seconds) Velocity 𝑣𝑣 (metres/second) 0 0 1 5.2004253 2 5.663451 3 6.1644492 4 6.7181163 5 7.330013 6 8.0062633 7 8.7536356 8 9.5796096 9 10.492452 10 11.501299 11 12.616247

The velocity values are realistic for Competitor 2. To verify the displacement model for Competitor 2, I substituted the parameter values to generate a refined displacement function: 𝑑𝑑Competitor 2 = 0.9𝑏𝑏 + 39𝑒𝑒0.1𝑓𝑓 + .774194𝑒𝑒−6.2𝑓𝑓 + 𝑝𝑝 Using 𝑝𝑝 = −39.7741935484 produced an appropriate representation of the displacement of Competitor 2 (see Table 3).

Table 3 Time 𝑏𝑏 (seconds) Displacement 𝑑𝑑 (metres) 0 –3.5484× 10−6 1 4.2290434 2 9.6605172 3 15.5703 4 22.00697 5 29.025936 6 36.68844 7 45.062162 8 54.221903 9 64.250328 10 75.238798 11 87.288281 12 100.51037

Note: The 𝑏𝑏 = 0 value is not exactly 0, but is considered accurate enough for the purpose of this report.



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Solve [4–5] application of mathematical concepts and techniques relevant to the task Shows understanding of calculus concepts and principles relevant to an aspect of the problem.

valuate and verify [2–3] statement about the reasonableness of solutions by considering the context … Writer has considered the reasonableness of the solution to Competitor 2 model.

Solve [6–7] accurate and appropriate use of technology Use of slider provides an efficient means of investigating function shape by changing parameter values (previous evidence in the response identifies writer’s expertise using the technology).

The velocity function was differentiated to determine acceleration at different times for Competitor 2. 𝑎𝑎competitor 2 = .39𝑒𝑒0.1𝑓𝑓 + 29.76𝑒𝑒−6.2𝑓𝑓 Table 4 summarises the findings.

Table 4 Time 𝑏𝑏 (seconds) Acceleration 𝑎𝑎 (metres/second/second) 0 30.15 1 0.4914 2 0.4765 3 0.5264 4 0.5818 5 0.6430 6 0.7106 7 0.7854 8 0.868 9 0.9592 10 1.0601 11 1.1716 12 1.2949

Competitor 2 is seen to be accelerating at time 𝑏𝑏 = 0 (where a standing start was assumed), accelerates at a decreasing rate initially and then accelerates at an increasing rate.

Displacement, velocity and acceleration models for Competitors 1 and 3 Using the Desmos graphics calculator and sliders, parameter values were found and the following models were generated (see Table 5): 𝒅𝒅𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏𝒔𝒔𝒔𝒔𝒔𝒔(.𝟏𝟏𝟎𝟎𝒕𝒕)

𝒅𝒅𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 𝟑𝟑 =𝟖𝟖𝒕𝒕

𝟏𝟏+ 𝒆𝒆−𝟑𝟑𝒕𝒕−𝟏𝟏𝟏𝟏

Table 5 Time 𝑏𝑏 (seconds)

Competitor 1 displacement 𝑑𝑑 (metres)

Competitor 3 displacement 𝑑𝑑 (metres)

0 0 0 1 9.8866404 7.999982 2 19.693253 15.999998 3 29.340458 23.99999 4 38.750166 31.99999 5 47.846209 39.99999 6 56.554959 47.99999 7 64.805293 56



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Solve [4–5] application of mathematical concepts and techniques … Shows understanding of calculus concepts and principles relevant to an aspect of the problem. Solve [2–3] use of simple procedures to make some progress towards a solution … Appropriate sequence to find the derivative. Formulate [3–4] accurate translation of … aspects of the problem …

8 72.532314 64 9 79.671589 72 10 86.16596 80 11 91.962858 88 12 97.015359 96 13 101.28257 104

The first and second derivatives for the displacement function were then found to determine the velocity and acceleration functions for Competitors 1 and 3.

𝑣𝑣𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝𝑒𝑒𝑏𝑏𝑐𝑐𝑏𝑏𝑐𝑐𝑐𝑐 1 =𝑑𝑑𝑠𝑠𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝𝑒𝑒𝑏𝑏𝑐𝑐𝑏𝑏𝑐𝑐𝑐𝑐 1

𝑑𝑑𝑏𝑏= 110 cos(.09𝑏𝑏) × .09 = 9.9 cos(.09𝑏𝑏)

𝑎𝑎𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝𝑒𝑒𝑏𝑏𝑐𝑐𝑏𝑏𝑐𝑐𝑐𝑐 1 =𝑑𝑑𝑣𝑣𝑐𝑐𝑐𝑐𝑚𝑚𝑝𝑝𝑒𝑒𝑏𝑏𝑐𝑐𝑏𝑏𝑐𝑐𝑐𝑐 1

𝑑𝑑𝑏𝑏= −9.9 sin(.09𝑏𝑏)× .09 = −.891 sin(.09𝑏𝑏)


𝑑𝑑𝑏𝑏

Using the quotient rule:

𝑣𝑣𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑐𝑐 3 =(1 + 𝑒𝑒−3𝑓𝑓−10) × 8 − 8𝑏𝑏 × −3 × (𝑒𝑒−3𝑓𝑓−10)

(1 + 𝑒𝑒−3𝑓𝑓−10)2

=8(1 + 𝑒𝑒−3𝑓𝑓−10) + 24𝑏𝑏(𝑒𝑒−3𝑓𝑓−10)

(1 + 𝑒𝑒−3𝑓𝑓−10)2

=8 + 8𝑒𝑒−3𝑓𝑓−10 + 24𝑏𝑏𝑒𝑒−3𝑓𝑓−10

(1 + 𝑒𝑒−3𝑓𝑓−10)2

Using the quotient, chain and product rules, the acceleration of Competitor 3 (𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑐𝑐 3) is: (1 + 𝑒𝑒−3𝑓𝑓−10)2 × (−24𝑒𝑒−3𝑓𝑓−10 + 24𝑏𝑏 ×−3𝑒𝑒−3𝑓𝑓−10 + 𝑒𝑒−3𝑓𝑓−10 × 24) − (8 + 8𝑒𝑒−3𝑓𝑓−10 + 24𝑏𝑏𝑒𝑒−3𝑓𝑓−10) × 2(1 + 𝑒𝑒−3𝑓𝑓−10) ×−3𝑒𝑒−3𝑓𝑓−10

(1 + 𝑒𝑒−3𝑓𝑓−10)4

Table 6 lists the value of these functions for the competitors.

Table 6: Competitor 1 Time 𝑏𝑏 (seconds)

Velocity 𝑣𝑣 (metres/second)

Acceleration 𝑎𝑎 (metres/second/second)

0 9.9 0 1 9.85993 –.080082 2 9.74005 –.159515 3 9.54133 –.237658 4 9.26538 –.313876 5 8.91443 –.387554 6 8.49132 –.458095 7 7.99947 –.524928 8 7.44288 –.587512 9 6.82603 –.64534 10 6.15394 –.697944 11 5.43203 –.744899 12 4.66615 –.785824 13 3.8625 –.820389



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Solve [6–7] accurate and appropriate use of technology Technology used to establish a best-fit relationship model.

Table 7: Competitor 3 Time 𝑏𝑏 (seconds)

Velocity 𝑣𝑣 (metres/second)

Acceleration 𝑎𝑎 (metres/second/second)

0 7.9996 0.002179 1 8.000036 –.000054247 2 8.0000045 –.000010803 3 8.0000003585 –.0000009413 4 8.0000000245 –.00000006695 5 8.0000000015 –.000000004333 6 8 –.0000000002655 7 8 –.0000000000157 8 8 –.0000000000009049 9 8 –.000000000000051198 10 8 –.000000000000002855 11 8 –1.57×10−16 12 8 –8.59×10−18 13 8 –4.656×10−19

Given data — developing a model for Competitor 4 Using the given data, technology was used to fit regression models. The models were refined using the 𝑅𝑅2 value (the closer this value is to 1, the better the ‘fit’ of the model).

Competitor 4 — 10-metre splits (linear)

Competitor 4 — 10-metre splits (quadratic)

y = 11.014x – 8.3767R² = 0.9888

-200

20406080

100120

0 2 4 6 8 10 12

Posi

tion

(met

res)

Time (seconds)

y = 0.3612x2 + 7.4362x – 2.5031R² = 0.9974

-200

20406080

100120

0 2 4 6 8 10 12

Posi

tion

(met

res)

Time (seconds)



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Evaluate and verify [2–3] statement about relevant strengths and limitations … Writer has determined the strength of the Competitor 4 model by considering the R2 value. Solve [2–3] use of simple procedures to make some progress towards a solution … Appropriate procedure identified. Evaluate and verify [2–3] statement about the reasonableness of solutions … Statement only about the appropriateness of acceleration and velocity values; no attempt has been made to use this information to refine the model. Evaluate and verify [2–3] statement about relevant strengths and limitations of the model Writer has verified the final place in the race for all competitors (given information in

Competitor 4 — 10-metre splits (cubic)

The cubic model had the highest 𝑅𝑅2 value. Using rules for differentiating polynomials, the velocity and acceleration functions for Competitor 4 were found:


𝑑𝑑𝑏𝑏= −0.2172𝑥𝑥2 + 2.8012𝑥𝑥 + 3.6516

𝑎𝑎𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑐𝑐 4 =𝑑𝑑𝑣𝑣𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑒𝑒𝑓𝑓𝑐𝑐𝑓𝑓𝑐𝑐𝑐𝑐 1

𝑑𝑑𝑏𝑏= −0.4344𝑥𝑥 + 2.8012

Table 8 lists the value of these functions for Competitor 4. Competitor 4

Table 8 Time 𝑏𝑏 (seconds) Velocity 𝑣𝑣

(metres/second) Acceleration 𝑎𝑎 (metres/second/second)

0 3.6516 2.8012 1 6.2356 2.3668 2 8.3852 1.9324 3 10.1004 1.498 4 11.3812 1.0636 5 12.2276 .6292 6 12.6396 .1948 7 12.6172 –.2396 8 12.1604 –.674 9 11.2692 –1.1084 10 9.9436 –1.5428

It is apparent that many of the values in the tables above are not appropriate, e.g. non-zero values for initial velocity and non-zero initial acceleration values. Table 9 summarises the findings and verifies that the positions assigned to each competitor are consistent with the developed models.

y = -0.0724x3 + 1.4006x2 + 3.6516x – 0.3663R² = 0.9998

-20

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Posi

tion

(met

res)

Time (seconds)



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the task).

Table 9 Competitor Function that models race Time at

position 100 m

Place in race

4 𝑑𝑑 = −.0724𝑏𝑏3 + 1.4006𝑏𝑏2 + 3.6516𝑏𝑏− .3663

9.6273 1

2 𝑑𝑑 = 0.9𝑏𝑏 + 39𝑒𝑒0.1𝑓𝑓 + .774194𝑒𝑒−6.2𝑓𝑓 −39.7741

11.9631 2

3 𝑑𝑑 =8𝑏𝑏

(1 + 𝑒𝑒(−3𝑓𝑓−10)) 12.5 3

1 𝑑𝑑 = 110 sin(0.09𝑏𝑏) 12.6789 4 The four models are graphed below:

Key: Orange — Competitor 4, Purple — Competitor 2, Green — Competitor 3, Blue — Competitor 1

Conclusion It is apparent from the analysis that none of the given or the developed models can accurately model the running of a 100-metre race. The displacement, velocity and acceleration models are generated using differentiation techniques, and this relationship made it problematic to find a model that could be used to determine the position, velocity and acceleration of any competitor at any time.

Time (seconds)

Communicate The conclusion should summarise 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. The writer has constructed an adequate conclusion only.

Solve The response has not included when and where the competitors were running the fastest and slowest, and when and where the competitors were accelerating the most and the least, as required in the task.

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Mathematical Methods 2020 v1...200041 Mathematical Methods 2020 v1.2 IA1 mid-level annotated sample response February 2020 Problem-solving and modelling task (20%) This sample has