mathematical methods 201 9 v1 - queensland curriculum and ......• correct use of appropriate...

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Mathematical Methods 2019 v1.2 IA1 high-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 2019 v1.2 IA1 high-level annotated sample response

Queensland Curriculum & Assessment Authority February 2020

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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 4

Solve Assessment objectives 1, 6 7 7

Evaluate and verify Assessment objectives 4, 5 5 4

Communicate Assessment objective 3 4 4

Total 20 19

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 documentation of relevant observations Formulate [3–4] documentation of appropriate assumptions

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 Place Competitor 1: 𝑑𝑑 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎(𝑏𝑏𝑏𝑏) 4th Competitor 2: 𝑣𝑣 = 𝑐𝑐(1− 𝑒𝑒−𝑓𝑓𝑓𝑓) + 𝑔𝑔(1− 𝑒𝑒ℎ𝑓𝑓) 2nd

Competitor 3: 𝑑𝑑 = 𝑗𝑗𝑓𝑓(1+𝑒𝑒−𝑘𝑘𝑘𝑘+𝑙𝑙)

+𝑚𝑚 3rd

Competitor 4: equation modelled from data 1st

where 𝑑𝑑 is the position in metres, 𝑏𝑏 is the time in seconds and 𝑣𝑣 is the velocity in metres per second.

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



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Communicate [3–4] 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 Solve [6–7] accurate and appropriate use of technology Communicate [3–4] correct use of appropriate technical vocabulary, procedural vocabulary, and conventions to develop the response

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 17.88 metres/second) (Live Science 2010). Therefore, it is assumed that no sprinter can exceed 17.88 metres/second.

4. Competitors use blocks to begin the race and accelerate from them.

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 𝑏𝑏 when the acceleration equals 0 Max./min. acceleration solve for 𝑏𝑏 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 (𝑏𝑏) values greater than or equal to zero and less than 11 seconds (0 ≤ 𝑏𝑏 ≤ 11).

Determining the models Competitor 1: 𝒅𝒅𝟏𝟏(𝒕𝒕) = 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 (𝒃𝒃𝒕𝒕) 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: Graph of data values

020406080

100120

0 2 4 6 8 10

Dis

tanc

e (m

)

Time (s)

Competitor 1



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

Communicate [3–4] coherent and concise organisation of the response …

Solve [6–7] accurate and appropriate use of technology

To determine the exact time Competitor 1 crosses the finish, the function was equated to 100 and the equation solved. 𝑑𝑑1(𝑏𝑏) = 157 sin(0.069𝑏𝑏)

100 = 157sin (0.069𝑏𝑏)

0.64 = sin(0.069𝑏𝑏)

𝑎𝑎𝑎𝑎𝑎𝑎−1(0.64) = 0.069𝑏𝑏

𝑎𝑎𝑎𝑎𝑎𝑎−1(0.64)0.069 = 𝑏𝑏

𝑏𝑏 = 10.07 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. 𝑣𝑣1(𝑏𝑏) = 10.833cos (0.069𝑏𝑏) and 𝑎𝑎1(𝑏𝑏) = −0.747sin (0.069𝑏𝑏) where 𝑣𝑣1(𝑏𝑏) and 𝑎𝑎1(𝑏𝑏) denote velocity and acceleration for Competitor 1. The table of values shows the velocity and acceleration at time 𝑏𝑏.

Time 𝑏𝑏 (seconds)

Velocity (m/s) Acceleration (m/s2)

0 10.833 0 1 10.8072 –.0515 2 10.73 –.1028 3 10.6017 –.1535 4 10.423 –.2036 5 10.1947 –.2526 6 9.9178 –.3004 7 9.5938 –.3469 8 9.224 –.3917 9 8.8105 –.4346 10 8.3549 –.4754 11 7.8596 –.5140

The times for the maximum and minimum velocity and acceleration are:

Velocity Acceleration 𝑏𝑏 (sec) 𝑣𝑣 (m/s) 𝑏𝑏 (sec) 𝑎𝑎 (m/s2) Maximum 0 10.833 0 0 Minimum 10.07 8.322 10.07 –0.478

These values were found by graphing the functions and analysing the graph to determine the global maximum and minimum.



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Solve [6–7] discerning application of mathematical concepts and techniques relevant to the task Writer has recognised the appropriate technique to use to solve.

Competitor 2: 𝒗𝒗 = 𝒄𝒄(𝟏𝟏− 𝒆𝒆−𝒇𝒇𝒕𝒕) +𝒈𝒈(𝟏𝟏 − 𝒆𝒆𝒉𝒉𝒕𝒕) The model given for Competitor 2 is a velocity function; therefore, the integral must be found to model displacement.

𝑑𝑑2(𝑏𝑏) = �𝑣𝑣𝑑𝑑𝑏𝑏

𝑑𝑑2(𝑏𝑏) = �𝑐𝑐(1− 𝑒𝑒−𝑓𝑓𝑓𝑓) + 𝑔𝑔(1− 𝑒𝑒ℎ𝑓𝑓) 𝑑𝑑𝑏𝑏

𝑑𝑑2(𝑏𝑏) = �𝑐𝑐 − 𝑐𝑐𝑒𝑒−𝑓𝑓𝑓𝑓 + 𝑔𝑔 − 𝑔𝑔𝑒𝑒ℎ𝑓𝑓𝑑𝑑𝑏𝑏

𝑑𝑑2(𝑏𝑏) = 𝑐𝑐𝑏𝑏 −𝑐𝑐𝑒𝑒−𝑓𝑓𝑓𝑓

−𝑓𝑓+ 𝑔𝑔𝑏𝑏 −

𝑔𝑔𝑒𝑒ℎ𝑓𝑓

ℎ+ 𝑎𝑎

𝑑𝑑2(𝑏𝑏) = �𝑐𝑐𝑏𝑏 + 𝑐𝑐𝑒𝑒−𝑓𝑓𝑘𝑘

𝑓𝑓�+ �𝑔𝑔𝑏𝑏 − 𝑔𝑔𝑒𝑒ℎ𝑘𝑘

ℎ�+ 𝑎𝑎 where 𝑎𝑎 is the constant of integration.

Using Desmos produced the parameter values 𝑐𝑐 = 10.5, 𝑓𝑓 = 10, 𝑔𝑔 = 0.105, ℎ = 0.1 and 𝑎𝑎 = 0

The displacement–time graph and table of values for this function are below:

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 𝑣𝑣2(𝑏𝑏) = 10.5(1− 𝑒𝑒−10𝑓𝑓) + 0.105(1− 𝑒𝑒0.1𝑓𝑓) and 𝑎𝑎2(𝑏𝑏) = 105𝑒𝑒−10𝑓𝑓 − 0.0105𝑒𝑒0.1𝑓𝑓

At time zero, the velocity is zero, which would be the case in a sprinting race.

0

20

40

60

80

100

120

-1 1 3 5 7 9 11

Dis

tanc

e (m

)

Time (s)

Competitor 2



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The table of values for both velocity and acceleration is:

𝑣𝑣2(𝑏𝑏) 𝑎𝑎2(𝑏𝑏)

The times for the maximum and minimum velocity and acceleration are:

Velocity Acceleration 𝑏𝑏 (sec) 𝑣𝑣 (m/s) 𝑏𝑏 (sec) 𝑎𝑎 (m/s2) Maximum 1 10.488 0 104.99 Minimum 9.69 10.328 9.69 –0.028

Note that at time zero, the sprinter has the greatest acceleration when leaving the blocks. Competitor 3: 𝑑𝑑 = 𝑗𝑗𝑓𝑓

(1+𝑒𝑒−𝑘𝑘𝑘𝑘+𝑙𝑙)+𝑚𝑚

The model for Competitor 3 is a logistical equation. Using Desmos, the following function was generated: 𝑑𝑑3(𝑏𝑏) = 10.3𝑓𝑓

1+𝑒𝑒(−0.47𝑘𝑘+1) The displacement at certain times and graph of the function are given below:

0

20

40

60

80

100

120

140

-1 1 3 5 7 9 11

Dis

tanc

e (m

)

Time (s)

Competitor 3



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Communicate [3–4] correct use of appropriate technical vocabulary, procedural vocabulary, and conventions to develop the response Calculus notation and equality signs used appropriately. Solve [6–7] 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

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.

𝑣𝑣3(𝑏𝑏) =10.3�1 + 𝑒𝑒(−0.47𝑓𝑓+1)� − 10.3t(−0.47𝑒𝑒(−.47𝑓𝑓+1))

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2

Simplifying by expanding and factorising:

𝑣𝑣3(𝑏𝑏) = 10.3+10.3𝑒𝑒(−0.47𝑘𝑘+1)+4.841𝑓𝑓𝑒𝑒(−.47𝑘𝑘+1)

(1+𝑒𝑒(−0.47𝑘𝑘+1))2=10.3�1+𝑒𝑒(−0.47𝑘𝑘+1)�+4.841𝑓𝑓𝑒𝑒(−.47𝑘𝑘+1)

(1+𝑒𝑒(−0.47𝑘𝑘+1))2= 10.3

�1+𝑒𝑒(−0.47𝑘𝑘+1)�+

4.841𝑓𝑓𝑒𝑒(−.47𝑘𝑘+1)

(1+𝑒𝑒(−0.47𝑘𝑘+1))2

To determine the function for acceleration:

Let 𝑤𝑤 = (4.841𝑓𝑓𝑒𝑒(−0.47𝑘𝑘+1))(1+𝑒𝑒(−0.47𝑘𝑘+1))2

and 𝑧𝑧 = 10.3(1+𝑒𝑒(−0.47𝑘𝑘+1))

and differentiate separately

𝑤𝑤 = (4.841𝑓𝑓𝑒𝑒(−0.47𝑘𝑘+1))(1+𝑒𝑒(−0.47𝑘𝑘+1))2

using product rule 𝑤𝑤′ = 𝑝𝑝𝑞𝑞′ + 𝑞𝑞𝑝𝑝′ 𝑝𝑝 = (4.841𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1)) 𝑞𝑞 = �1 + 𝑒𝑒(−0.47𝑓𝑓+1)�−2 𝑝𝑝′ = −2.275𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1) + 4.841𝑒𝑒(−0.47𝑓𝑓+1) 𝑞𝑞′ = 0.94𝑒𝑒(−0.47𝑓𝑓+1)(1 + 𝑒𝑒(−0.47𝑓𝑓+1))−3 𝑤𝑤′ = �4.841𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1)� ∙ 0.94𝑒𝑒(−0.47𝑓𝑓+1)�1 + 𝑒𝑒(−0.47𝑓𝑓+1)�

−3 + �1 + 𝑒𝑒(−0.47𝑓𝑓+1)�−2

∙ −2.275𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1) + 4.841𝑒𝑒(−0.47𝑓𝑓+1)

𝑤𝑤′ =4.56𝑏𝑏𝑒𝑒2(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))3+−2.275𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1) + 4.841𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2

𝑧𝑧 = 10.3

(1+𝑒𝑒(−0.47𝑘𝑘+1)) using product rule 𝑧𝑧′ = 𝑟𝑟𝑎𝑎′ + 𝑎𝑎𝑟𝑟′

𝑟𝑟 = 10.3 𝑎𝑎 = (1 + 𝑒𝑒(−0.47𝑓𝑓+1))−1 𝑟𝑟′ = 0 𝑎𝑎′ = 0.47𝑒𝑒(−0.47𝑓𝑓+1)(1 + 𝑒𝑒(−0.47𝑓𝑓+1))−2

𝑧𝑧′ =4.841𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2

Combine parts 𝑤𝑤′ and 𝑧𝑧′

𝑎𝑎3(𝑏𝑏) =4.56𝑏𝑏𝑒𝑒2(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))3 +−2.275𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1) + 4.841𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2 +4.841𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2

𝑎𝑎3(𝑏𝑏) =4.56𝑏𝑏𝑒𝑒2(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))3 +−2.275𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1) + 9.682𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2

𝑎𝑎3(𝑏𝑏) =𝑒𝑒(−0.47𝑓𝑓+1)

(1 + 𝑒𝑒(−0.47𝑓𝑓+1))2 �4.56𝑏𝑏𝑒𝑒(−0.47𝑓𝑓+1)

1 + 𝑒𝑒(−0.47𝑓𝑓+1) − 2.275𝑏𝑏 + 9.682�



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Communicate [3–4] 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 Solve [6–7] accurate and appropriate use of technology Solve [6–7] discerning application of mathematical concepts … Writer has forced the initial values for position to zero and minimised error for finishing time. Evaluate and verify [4–5] evaluation of the reasonableness of solutions …

The table of values for both functions is: 𝑣𝑣3(𝑏𝑏) 𝑎𝑎3(𝑏𝑏)

Using these models:

Velocity Acceleration 𝑏𝑏 (sec) 𝑣𝑣 (m/s) 𝑏𝑏 (sec) 𝑎𝑎 (m/s2) Maximum 5.9489 12.349 1.6115 2.4943 Minimum 0 2.77 8.215 –.3576

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, 𝑅𝑅2 the more useful the model.

Polynomial Regression model (𝒅𝒅) 𝑹𝑹𝟐𝟐 Linear 9.7894𝑏𝑏 0.973 Quadratic 0.4342𝑏𝑏2 + 6.5209𝑏𝑏 0.9967 Cubic −.0747𝑏𝑏3 + 1.4423𝑏𝑏2 + 3.4221𝑏𝑏 0.9998 Fourth degree . 0092𝑏𝑏4 − .2507𝑏𝑏3 + 2.4803𝑏𝑏2 + 1.5896𝑏𝑏 1 Fifth degree −.0013𝑏𝑏5 + 0.0418𝑏𝑏4 − .5334𝑏𝑏3 + 3.4861𝑏𝑏2

+ .3829𝑏𝑏 1

Sixth degree . 0003𝑏𝑏6 − .0117𝑏𝑏5 + 0.1606𝑏𝑏4 − 1.1733𝑏𝑏3+ 5.0841𝑏𝑏2 − 1.0684𝑏𝑏

1

The 𝑅𝑅2 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 (�𝑠𝑠𝑎𝑎𝑎𝑎𝑘𝑘𝑎𝑎𝑎𝑎𝑙𝑙−𝑠𝑠𝑎𝑎𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑙𝑙�

𝑠𝑠𝑎𝑎𝑎𝑎𝑘𝑘𝑎𝑎𝑎𝑎𝑙𝑙× 100%), produced in a spreadsheet program, shows

nearly all values are less than 1% off.



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Solve [6–7] accurate and appropriate use of technology Solve [6–7] accurate use of complex procedures to reach a valid solution The solution consists of an involved combination of parts that are interconnected Solve [6–7] discerning application of mathematical concepts …

Appropriate formulas were used to determine 𝑎𝑎𝑢𝑢𝑠𝑠𝑢𝑢𝑢𝑢𝑔𝑔 𝑚𝑚𝑚𝑚𝑚𝑚𝑒𝑒𝑚𝑚, 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 10.28907156 value was generated using the equation = 0.0092*𝐴𝐴34 – 0.2507*𝐴𝐴33 + 2.4803*𝐴𝐴32 + 1.5896*𝐴𝐴3. Similarly, the percentage error value was generated using the equation �𝐵𝐵3−𝐷𝐷3

𝐵𝐵3�× 100.

Given: 𝑎𝑎4(𝑏𝑏) = .0092𝑏𝑏4 − .2507𝑏𝑏3 + 2.4803𝑏𝑏2 + 1.5896𝑏𝑏, using calculus methods:

𝑣𝑣4(𝑏𝑏) = .0368𝑏𝑏3 − .7521𝑏𝑏2 + 4.9606𝑏𝑏 + 1.5896 𝑎𝑎4(𝑏𝑏) = .1104𝑏𝑏2 − 1.5042𝑏𝑏 + 4.96 To find the local maximum and minimum velocities: 𝑣𝑣′4(𝑏𝑏) = 0. 0 = .1104𝑏𝑏2 − 1.5042𝑏𝑏 + 4.96 Using the quadratic formula:

𝑏𝑏 = 1.5042 ±�((−.15042)2 − 4 × .1104 × 4.96)2 × .1104

𝑏𝑏 ≈ 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: 𝑣𝑣4′(5) = .199 and 𝑣𝑣4′(7)− .1598 and 𝑣𝑣4′(. 805) = .1507 The gradient is positive to the left and negative to the right of 𝑏𝑏 = 5.59; therefore, a local maximum velocity occurs at time 𝑏𝑏 = 5.59 seconds and, using similar reasoning, a local minimum at time 𝑏𝑏 = 8.03 seconds. However, it is clear from the graph of the velocity model that the maximum velocity occurs at the end of the race at 𝑣𝑣4(9.59) = 12.4492 metres/second, and the minimum value occurs at the start 𝑣𝑣4(0) = 1.5896 metres/second.



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Evaluate and verify [4–5] documentation of relevant strengths and limitations of the solution and/or model


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 𝑏𝑏 (sec) 𝑣𝑣 (m/s) 𝑏𝑏 (sec) 𝑎𝑎 (m/s2) Maximum 9.59 12.4492 0 4.96 Minimum 0 1.5896 6.8125 –.1631

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.

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 10.488 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 10.328 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.



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Communicate [3–4] 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.

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.

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’,

www.livescience.com/8039-humans-run-40-mph-theory.html. Olympic Games 2017, ‘100m Men’, www.olympic.org/athletics/100m-men.

http://www.livescience.com/8039-humans-run-40-mph-theory.html

http://www.olympic.org/athletics/100m-men

mathematical methods 201 9 v1 - queensland curriculum and ......• correct use of appropriate...

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