Applications and Modelling

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Mathematics Advanced Year 11 Sample Assessment Task

Applications and Modelling

Sample for implementation for Year 11 from 2019

Context

Prior to undertaking this assignment students will have engaged in learning for multiple topics.

These include:

 MA-F1 Working with Functions

 MA-T2 Trigonometric Functions and Identities

 MA-C1 Introduction to Differentiation

 MA-E1 Logarithms and Exponentials

 MA-S1 Probability and Discrete Probability Distributions

Students will have participated in activities to develop knowledge of the concepts within these topics, and the skills to solve a variety of problems.

Part of the assignment involves the exploration of concepts that form part of the Mathematics Advanced Year 12 course. It is not necessary for the students to have covered this content prior to undertaking the task.

Students will require approximately six hours of independent preparation and at least one session of class time. Teachers are advised to allocate some additional time during class to discuss the notification and task requirements.

Notes to teacher

Throughout the development of the assignment, teachers should monitor authorship and student progress. All assigned responses and tasks must be submitted on the same day.

For Investigation 1 of the assignment it is appropriate for the students to work in pairs or small groups to streamline the collection of data. Other suggestions for modifying the assignment appear later in this document.

The marking criteria are provided within the task notification. Teachers are advised to allocate time to discuss the marking criteria with students to highlight the scope of the task and clarify expectations.

The marking guidelines provided at the end of this document illustrate an approach for how marks may be allocated to student responses.

When feedback is provided after marking, there will be opportunity to discuss the challenges of the task with the class and consider future learning activities to assist student learning. It is appropriate for the teacher to discuss the marking guidelines with students as part of the feedback provided upon completion of the task.

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Applications and Modelling

Task number: 3 Weighting: 25% Timing: Term 3, Week 6 Outcomes assessed

˃ uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems MA11-1

˃ uses appropriate technology to investigate, organise, model and interpret information in a range of contexts MA11-8

˃ provides reasoning to support conclusions which are appropriate to the context MA11-9 Nature of the task

This investigative assignment involves the collection of data and the use of functions to create mathematical models. You will be allocated one lesson of class time and should plan to spend approximately 6 hours of individual work.

You can choose to work with a partner or small group for Investigation 1. Each member of the group will be awarded the same mark for the section. You must provide the names of all contributors and indicate the section of the work that was completed collaboratively.

All graphs, tables and outputs generated in your investigations should be clearly and concisely explained in a report that is to be handed in by 3:30 pm on the due date. There is no set format for the presentation of your report but evidence of your work with technology must be included.

Digital files must be submitted. You may also submit your report in digital form.

Authentication

In order to verify that the work you submit is your own:

 Place a digital signature comprised of your name and date of birth in the top left corner of all digital files.

 Ensure that this digital signature is clearly visible in any screenshot included with your submission.

Failure to authenticate your work in this way may be interpreted as a non-attempt.

Marking criteria

You will be assessed on how well you:

 use algebraic and graphical techniques to solve familiar and unfamiliar problems, comparing alternative solutions where appropriate

 use appropriate technology to investigate, organise, model and interpret information

 apply your knowledge of functions and relations, logarithmic and exponential functions, statistical analysis and probability, to solve problems

 communicate your processes and findings

 provide reasoning and justification to support conclusions.

Feedback provided

 After your assignment has been marked you will be provided with feedback outlining your strengths and indicating areas for improvement so that you can build your knowledge, skills and understanding and plan future learning.

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Applications and Modelling – Investigations

Student Resources

Generating random numbers

Most spreadsheet and calculator functions create pseudo-random numbers because they use a formula to generate the numbers. True random numbers are created using a random phenomenon such as atmospheric noise.

Working with Excel

To generate a random number use:

 =RAND() creates a number between 0 and 1

 =RAND()*N creates a number between 0 and N

 =RAND()*(B-A)+A creates a number between A and B but not including B

 =INT(RAND()*N) creates an integer between 0 and N

 =INT(RANDBETWEEN(A,B)) creates an integer between A and B To split the digits of a number into separate cells use:

 =MID($A1,COLUMN()-1,1) where $A1 is the location of the first number you need to split. This will split off the first digit.

 Copy the formula both down and to the right to complete the separation.

 For an explanation of how this process works see:

http://www.exceltip.com/excel-text-formulas/separating-a-number-into-digits.html To count how many digits there are in a column use:

 =COUNTIF(RANGE,D) where D is the digit you desire and the range is expressed as, for example, $A$1:$A$200

Domesday data

The Domesday Book (pronounced ‘doomsday’) is a document that was created in England in 1086. It records data from a survey of over 13 000 settlements. The data was originally collected to assess wealth in order to determine taxes. It records land use, asset value and local information that give a snapshot of life in 11th-century England.

The information within the Domesday Book is available on line at https://opendomesday.org.

Entering the name of a town that was in existence in 1066 will reveal the details that were recorded. For example, entering ‘London’ into the search cell labelled ‘Find Domesday places near:’ will produce a list of towns in the vicinity of London. Selecting Westminster, the first of these, reveals the entries that were recorded. Other town names can be found using the map that appears on the site.

Moon and tide information

Moon distances from Earth: www.timeanddate.com

Australian tide information and graphs of tides: http://willyweather.com.au

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Investigation 1 – Random Numbers (5 marks)

This investigation explores the leading digits of random numbers using an Excel spreadsheet.

You may choose to work with a partner or small team for this investigation. Record the names of each collaborator and submit the list with your response to the investigation.

1. Using Sheet 1 of an Excel spreadsheet file:

 generate a set of 450 random integers from 1 to 999

 produce a column graph that illustrates the frequency of the leading digit of the numbers.

2. Using Sheet 2 of the spreadsheet file, repeat this process using true random integers generated at www.random.org.

3. (a) Using the Domesday Book data:

 find and record 450 data points in Sheet 3 of the spreadsheet file

 include a column which records the name of each town you have used to collect your data

 do not include any data that relates to the year in which the information was collected

 record any decimals as integers, ignoring the decimal point.

For example, using the entry for Battersea:

Source: http://opendomesday.org/place/TQ2676/battersea/

(b) Produce a column graph that illustrates the frequency of the leading digit of the numbers.

4. Compare and contrast the three column graphs created so far. A copy of each column graph must be included in the body of your response.

5. Submit your digital files for this investigation as Investigation1_yourname.filetype.

Batterse a

70 72 72 180 605 80 755 30 3 14 82 50 7 424 8

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Investigation 2 – Benford’s Law (15 marks)

1. (a) Using graphing software create a sketch of the function y=log_e

(

^x⁺¹^x

)

^.

(b) With reference to the algebra of logarithms, give reasons for the shape of the graph.

2. (a) Using graphing software, plot the Domesay Book data you collected for Investigation 1 onto a Cartesian number plane.

(b) Find a function that will best fit the data. Clearly explain your strategy and state any limitations on your model.

3. Research Benford’s Law. In approximately 200 words, explain what it is and how it is used.

4. (a) Write three questions related to Benford’s Law that you could explore mathematically.

(b) Investigate the answer to one of your questions, providing evidence to support your conclusion.

Investigation 3 – Logistics (15 marks)

1. The following table shows worldwide unit sales of the Apple iPhone from 2007 to 2017.

Year Unit sales in millions Year Unit sales in millions

2007 1.39 2013 150.26

2008 11.63 2014 169.22

2009 20.73 2015 231.22

2010 39.99 2016 211.88

2011 72.29 2017 216.76

2012 125.05

Source: https://www.statista.com/statistics/276306/global-apple-iphone-sales-since-fiscal-year-2007/

(a) Using graphing software, plot only the data for 2007 to 2012 inclusive. Find a function that will best fit the data. Clearly explain your strategy and state any limitations on your model.

(b) Use your model to predict the unit sales in millions for the year 2014. Comment on the validity of your prediction without reference to the data in the table.

2. (a) Using graphing software and creating sliders for k, a and b, sketch the graph of the logistic function f(x)= k

1+e^−a⁽^x+b⁾^.

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(b) With reference to your knowledge of the gradient function, explain the effect that changing each of the values of k, a and b has on the graph.

3. (a) Now plot all available data and fit a better model to the data using the logistic function. Clearly explain your strategy and state any limitations on this new model.

(b) Use your model to predict the unit sales in millions for 2018. Comment on the validity of your prediction. What advice about future sales could you give to the manufacturer?

4. Submit your digital files for this investigation as Investigation3_yourname.filetype.

Investigation 4 – The Changing Tide (10 marks)

1. (a) Using graphing software and creating sliders for k, a and b, sketch the graph of the function f(x)=ksina(x+b)^.

(b) Explain the effect that changing each of the values of k, a and b has on the graph.

2. (a) Add the function g(x)=KsinA(x+B) to the file you have created for Question 1, including additional sliders for K, A and B. Using both of these graphs, produce the graph of h(x)=f(x)+g(x). Explore the effect of changing the values of k, a, b, K, A and B on the graph of h(x)^.

(b) Make three observations that result from your investigation.

3. (a) Perigee is a point in the moon’s orbit when it is closest to the Earth. The date of one perigee in 2017 was 18 August 2017. Using the site www.timeanddate.com, find the date of a perigee to investigate.

(b) Use the resources on the site http://willyweather.com.au to plot the graph of the tides for the date you have chosen. Take a snapshot of the graph.

4. (a) Using digital technology, find a curve that models the tides for the date you have chosen. What are the strengths and limitations of your model?

(b) Using your model, predict the height of the tide at 6 am on the day following perigee.

End of assignment

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Teacher Resources

Student responses may include the submission of digital files. Teachers are advised to determine a file-naming protocol that students use consistently. This will help streamline the process of accessing the digital resources associated with each student’s work.

The investigations can be altered or individualised in the following ways:

Investigation 1

 Complete the collection of data for this investigation in class. For students who have limited experience in working with Excel, this will provide an opportunity for them to seek teacher assistance.

 If the students have opted to work in a small group, they could be required to assess the contribution each group member made to the group’s response. Students could then be allocated a proportion of the marks awarded to the response.

 The teacher is advised to keep a record of student groups in order to monitor authorship.

Investigation 2

 Provide the students with individualised, fraudulent financial data. Students explore the data and provide advice about which transactions should be questioned with reasons why they are questionable.

 Fraudulent datasets are available at: https://www.kaggle.com/ntnu-testimon/paysim1

 Provide each student with a different data source for which they check the validity of Benford’s Law. Note that this data should be ‘naturally occurring’. For example, students could be directed to use:

 a specific data source available online such as the webpage of an historical house

 documents found online

 a page from a local newspaper

 school inventory records.

Investigation 3

Other sources of real data can be found at:

 https://ourworldindata.org/public-spending

 https://ourworldindata.org/financing-education

 www.statista.com – Note that this website provides access to some basic statistics for free.

Investigation 4

 Students could select their birthdate for the previous 12-month period and explore the tides that occurred on that day. This would require the elimination of the references to perigee.

 Students explore polynomial graphs and fit a polynomial curve to the data. Note that they will then be unable to predict the tide for Question 4 (b).

References

Bellos, Alex, Alex Through the Looking-Glass, Bloomsbury, London, 2004

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Marking Guidelines

Investigation 1 – 5 marks

The student: Novice 0–1 mark Apprentice 2–3 marks Practitioner 4 marks Expert 5 marks uses appropriate

technology to investigate, organise, model and interpret information in a range of contexts MA11-8

There is some evidence that technology has been used to collect, organise and graph two sets of data.

There is an attempt to compare graphs.

Technology has been used to collect, organise and display three sets of data.

Similar aspects of the graphs are stated. Limited use of mathematical language evident in the explanations.

Technology has been used effectively to collect, organise and display all the required data. Formulae have been used for some calculations.

Formal mathematical language is used to describe similar aspects of the graphs and there is an attempt to contrast the graphs.

Technology has been used efficiently to collect, organise and display all the required data. Formulae have been used for all calculations.

Formal mathematical language is used to compare and contrast the graphs. All explanations are concise and correct.

The student: Novice 0–1 mark Apprentice 2–3 marks Practitioner 4 marks Expert 5 marks uses algebraic and

graphical

techniques to solve, and where

appropriate, compare alternative solutions to problems MA11-1

A graph for Question 1 has been partially created.

There is evidence of a basic understanding of the laws of logarithms and an attempt to explain one feature of the graph.

The correct graph for Question 1 has been created.

The function is written as a difference of logarithms.

There is an attempt to explain the domain restriction and one other feature of the graph.

The function is written as a difference of logarithms and this is used to describe the shape of the graph in the first quadrant.

The domain restriction is partially explained using formal mathematical language. At least one other feature of the graph is described precisely. There are no errors in the explanations.

The function is written as a difference of logarithms and this is used to describe the shape of the graph in the first quadrant.

There is acknowledgement that the difference of logarithms creates a function that is not identical to the original function.

There is an explanation for why the graph exists in the third quadrant.

The domain restriction is explained using formal mathematical language. All other features of the graph are described accurately.

There are no errors in the explanations.

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uses appropriate technology to investigate, organise, model and interpret information in a range of contexts MA11-8

There is some attempt to fit an exponential or logarithmic curve to the data.

There is an elementary explanation of

Benford’s Law that uses limited mathematical language.

The proposed questions for Question 4 (a) require minimal mathematical exploration.

An appropriate exponential or logarithmic curve has been fitted to the data but there is no evidence of an attempt to refine the fit.

The explanations of the strategy used and of Benford’s Law include some mathematical terminology.

An application of Benford’s Law is stated.

The proposed questions for Question 4 (a) are all appropriate and at least one requires an investigation of some depth.

There is significant evidence that the fitted curve has been refined in an attempt to improve the fit. The information in Question 1 has been used in this process.

The explanations of the strategy used and of Benford’s Law are clearly articulated using formal mathematical language and terminology. An application of Benford’s Law is explained.

All the proposed questions for Question 4 (a) are appropriate and require

comprehensive investigation.

There is extensive evidence that the fitted curve has been refined in an attempt to improve the fit. The information in Question 1 has been used in this process but the base has been altered

appropriately.

The explanations of the strategy used and of Benford’s Law are clearly articulated using formal mathematical language and terminology. A number of applications of Benford’s Law are explained.

All the proposed questions for Question 4 (a) reveal depth of insight and require comprehensive investigation.

provides reasoning to support

conclusions which are appropriate to the context MA11-9

There is evidence of an attempt to explore.

There is evidence of elementary explorations of a simple question and a conclusion has been made.

There is evidence of significant

exploration and a conclusion is stated in mathematical language.

Explorations are well designed and the process is explained clearly. The conclusion is articulated in formal mathematical language that includes correct use of terminology.

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There is some attempt to fit an appropriate curve to the data in Question 1.

An appropriate curve has been fitted to the data in Question 1.

There is an attempt explain the strategy used.

The model is used to predict future sales.

An appropriate curve has been fitted to the data in Question 1. The strategy used is clearly explained and includes a limitation of the model.

There is an elementary comment on the validity of the prediction.

The model fitted to the data in Question 1 is not exponential. There is evidence of an attempt to refine the model. The strategy used is clearly explained and includes a limitation of the model.

The validity of the prediction is expressed clearly in formal mathematical language.

uses algebraic and graphical

techniques to solve and where

appropriate, compare alternative solutions to problems MA11-1

There is an attempt to create the graph for Question 2.

There is an attempt to explain the effect of one variable on the shape the graph.

The graph for Question 2 has been created.

There is an attempt to explain the effect of two variables on the shape the graph.

There is an elementary reference to the gradient function.

The effect of each variable on the shape of the graph is explained using some mathematical language.

The gradient function is included in the sketch and the explanation refers to it.

There are no errors in the explanations.

The effect of each variable on the shape of the graph is expressed precisely in formal mathematical language.

The gradient function is included in the sketch and the explanation links the curve to its gradient function. There are no errors in the explanations.

There is some attempt to fit the logistic curve to the data.

The logistic curve has been fitted to the data. There is an attempt to explain the strategy used.

The model is used to predict future sales. There is no attempt to support the conclusion with reasoning.

The logistic curve has been fitted to the data. The strategy used is clearly explained and includes a limitation of the model.

The model is used to predict future sales. There is a comment on the validity of the prediction that is supported with elementary reasoning but there is no attempt to provide advice to the manufacturer.

The logistic curve has been fitted to the data. The strategy used is clearly explained in mathematical language and includes explanation of limitations of the model.

The model is used to predict future sales. The validity of the prediction is clearly articulated in mathematical language. The advice to the

manufacturer includes reference to other features of the data.

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There is an attempt to create the graphs for Questions 1 and 2.

There is an attempt to explain the effect of one variable on the shape the graph in Question 1.

Question 3 has been attempted.

The graphs for Questions 1 and 2 have been created.

There is an attempt to explain the effect of two variables on the shape of the graph in Question 1.

Some information for Question 3 is presented.

There is an attempt to model the tides.

The effect of each variable on the shape of the graph in Question 1 is explained using some mathematical language.

The information for Question 3 is clearly presented.

There is significant evidence that a model to fit the tides has been explored.

The effect of each variable on the shape of the graph in Question 1 is expressed precisely in formal mathematical language.

The information for Question 3 is clearly presented.

The model for the tides is a reasonable fit.

An elementary

observation is made on the relationships between the graphs in Question 2.

There is no evidence of explanations of strengths and

limitations of the model.

Three elementary observations are made on the relationships between the graphs in Question 2.

There is an elementary attempt to discuss the strength of the model in mathematical terms.

Three correct observations on the relationships between the graphs in Question 2 are expressed using mathematical language.

There is some discussion of the strengths and limitations of the model using mathematical language.

There is an attempt to predict the 6 am tide. The prediction does not have to be correct.

Three correct observations on the relationships between the graphs in Question 2 are expressed precisely using mathematical language and terminology.

Strengths and limitations of the model are clearly explained in formal

mathematical language with no errors in reasoning.

The prediction of the 6 am tide includes evidence of calculations involving the scale on each axis. The prediction does not have to be correct.