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YEAR 12 MATHEMATICS B Assignment
Assessment Item: 16
Date Issued: Mon 8 th Aug
Monitoring Date: Mon 22 nd Aug
Date Due: Mon 5th Sept
Time Allowed: 4 weeks
Topics Assessed: Integration
Assessment Type: Assignment
Instructions:
1. All questions require FULLY WORKED SOLUTIONS.
2. Calculators are allowed and graphing calculators can be used where appropriate.
3. Ensure the work submitted is your own work. Work that has been plagiarised will not be marked.
Submission Details:
1. The assignment should be submitted on A4 paper with the assignment sheet stapled to the front of your work with your name and teacher’s name written clearly on the front.
2. Assignments must be given to your teacher during the specified lessonon the due date.
3. Assignments can be submitted earlier than this date. Assignments received after the due date will receive a grade based on the work seen by your teacher prior to the due date through the submission of rough drafts. The grade awarded will only be approximate so an incomplete assignment handed in by the due date is a better option.
4. If illness prevents you from handing in your assignment on the due date, the school office must be contacted on the due day by phone. The assignment plus a doctor’s certificate must be given to the Head of Department on your return to school.
5. Do not ask your teacher for an extension, as they will be unable to grant your request. Occasionally, the Head of Department, Mrs Wiffen, may grant an extension. Extensions will only be granted if they are negotiated before the due date and the reason for the request is substantial.
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Mathematics B: Knowledge and procedures (Minimum Standards)
Attribute |
Standard A |
Standard B |
Standard C |
Standard D |
Standard E |
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recall, selection and use of definitions, results and rules |
· recalled and correctly used the definitions of - inverse of a function - relative area - percentage / proportion in both tasks · recalled and correctly used the trapezoidal rule in task 2 (c) (i) and (ii) |
· recalled and correctly used most of the following definitions - -inverse of a function - relative area - percentage / proportion in both tasks · recalled and correctly used the trapezoidal rule in task 2 (c) (i) or (ii) |
· recalled and correctly used the following definitions - inverse of a function - relative area - percentage / proportion in one task · recalled and correctly used the trapezoidal rule in task 2 (c) (i) or (ii |
· recalled and correctly used some of the following definitions - inverse of a function - relative area - percentage / proportion in one task.
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· rarely recalled and/or correctly used the following definitions - inverse of a function - relative area - percentage / proportion
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use of technology |
identified and used the most appropriate graphics calculator or software functions to · draw a representation of the tiles in tasks 1and 2 AND · calculate the relevant areas in tasks 1 and 2
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identified and used the graphics calculator or software functions to · draw a representation of the tiles in tasks 1and 2 AND · calculate the relevant areas in tasks 1 and 2 Some minor errors
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identified and used the graphics calculator or software functions to · draw a representation of the tiles in tasks 1and 2 OR · calculate the relevant areas in tasks 1 and 2
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identified and used the graphics calculator or software functions to · draw a representation of the tiles in task 1or 2 OR · calculate the relevant areas in task 1 or 2
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· demonstrated limited use of graphics calculator or computer software functions |
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selection and use of procedures
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accurately used · the correct procedure for calculation of the inverse of both functions · appropriate procedures for calculating the points of intersection of function and its inverse · the correct calculus approach to find the appropriate areas. · an appropriate procedure to calculate percentage of each colour · the trapezoidal rule to calculate area for 5 and 10 strips
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accurately used most of · the procedure for calculation of the inverse of both functions · procedures for calculating the points of intersection of function and its inverse · the correct calculus approach to find the appropriate areas. · an appropriate procedure to calculate percentage of each colour · the trapezoidal rule to calculate area for 5 and 10 strips
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accurately used some of · the procedure for calculation of the inverse of both functions · procedures for calculating the points of intersection of function and its inverse · the correct calculus approach to find the appropriate areas. · an appropriate procedure to calculate percentage of each colour · the trapezoidal rule to calculate area for 5 and 10 strips
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attempted to used some of · the procedure for calculation of the inverse of both functions · procedures for calculating the points of intersection of function and its inverse · the correct calculus approach to find the appropriate areas. · an appropriate procedure to calculate percentage of each colour · the trapezoidal rule to calculate area for 5 and 10 strips
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· attempted to use few of the listed procedures |
Mathematics B: Modelling and problem solving (Minimum Standards)
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Attribute |
Standard A |
Standard B |
Standard C |
Standard D |
Standard E |
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The student has: |
The student has: |
The student has: |
The student has: |
The student has: |
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Group A |
form a mathematical model of a life-related situation |
· developed a model for tasks 3, 4, and 5 |
· developed a model for tasks two of the tasks |
· developed a model for two of the tasks |
· developed a model for one of the tasks |
· developed a model for one of the tasks |
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derive results from consideration of the mathematical model |
· accurately use the model to determine the proportion of colour in tasks 3, 4 and 5 |
· accurately use the model to determine the proportion of colour in two of the tasks. |
· accurately use the model to determine the proportion of colour in two of the tasks |
· accurately use the model to determine the proportion of colour in one of the tasks |
· inaccurately used the model to determine the proportion of colour in the tasks |
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modify the model to fit the model to a given set of parameters |
· modified the models if necessary to fit the set parameters |
· modified two of the models if necessary to fit the set parameters |
· modified one of the models if necessary to fit the set parameters |
· modified one of the models if necessary to fit the set parameters |
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synthesis of procedures and strategies |
· accurately used appropriate procedures to demonstrate that the final models were consistent with the set parameters |
· accurately used appropriate procedures to demonstrate that the final two models were consistent with the set parameters |
· accurately used appropriate procedures to demonstrate that one of the final models was consistent with the set parameters |
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Attribute |
Standard A |
Standard B |
Standard C |
Standard D |
Standard E |
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The student has: |
The student has: |
The student has: |
The student has: |
The student has: |
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Group B – in some contexts & topics |
solving problems
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· solved two of the problems, giving solutions consistent with the set parameters allowing for minor errors. |
· solved one of the problems, giving a solution consistent with the parameters allowing for minor errors. |
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explored the strengths and limitations of a mathematical model |
· thoroughly discussed the abnormalities of the rule in task 5 and its limitation to n>1. |
· discussed the abnormalities of the rule in task 5 and its limitation to n>1. |
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Note: Shaded boxes are criteria above the minimum criteria for each of the standards and are to be used to allocate levels within each standard.
Mathematics B: Communication and justification ( Minimum standards)
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Attribute |
Standard A |
Standard B |
Standard C |
Standard D |
Standard E |
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use of mathematical terms and symbols |
· used appropriate terms for each situation. |
· used terms appropriately in most situations. |
· used basic terms appropriately in most situations. |
· used basic terms appropriately in some situations. |
· used basic terms appropriately in few situations. |
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· used symbols appropriately in each situation. |
· used symbols appropriately in most situations. |
· used basic symbols appropriately in most situations. |
· used basicsymbols appropriately in some situations. |
· used basicsymbols appropriately in few situations. |
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· used correct units for each situation. |
· used correct units for most situations. |
· used correct units for some situations. |
· used correct units for few situations. |
· used incorrect or no units. |
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use of language |
· consistently used accepted spelling , punctuation and sentence structure |
· generally used accepted spelling, punctuation and sentence structure . |
· used spelling, punctuation and sentence structure which enabled meaning to be conveyed inmost situations. |
· used spelling, punctuation and sentence structure which enabled meaning to be conveyed insome situations. |
· used spelling, punctuation and sentence structure which enabled meaning to be conveyed infew situations. |
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collection and organization of information |
· clearly labelled all question numbers and parts, setting out responses so that all responses can be easily followed. |
· clearlylabelled mostquestion numbers and parts, setting out most responses so that responses can be easily followed. |
· labelled mostquestion numbers and parts, setting out responses so that responses can be followed. |
· labelled mostquestion numbers , setting out responses so that some responses can be followed. |
· made some attempt to label questions and set out responses so that they can be followed. |
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· clearly framed responses to all items in terms of the original questions. |
· framed responses to most items in terms of the original questions |
· framed responses to allitems attempted in terms of the original questions. |
· framed responses to most items attempted in terms of the original questions. |
· rarely framed responses to items attempted in terms of the original questions. |
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· provided clear, accurate, labelled diagrams of the tiles and their tessellations for all tasks |
· provided clear, accurate, labelled diagrams of the tiles and their tessellations for most tasks |
· provided clear, accurate, labelled diagrams of the tiles and their tessellations for some tasks |
· provided labelled diagrams of the tiles for some tasks |
· provided a labelled diagram of the tiles for one of the tasks |
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use of mathematical reasoning |
· provided responses to all items, all of which are logical in their setting out, efficiently conveying the reasoning processes used. |
· provided responses to most items, most of which are logical in their setting out, conveying the reasoning processes used. |
· provided responses to items, most of which are logical in their setting out. |
· provided responses to items, some of which are logical in their setting out. |
· provided responses to items which are difficult to follow. |
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justification of procedures and conclusions |
· provided clear and concise evidence to justify the choice of model and modifications made to the model in all tasks |
· provided evidence to justify the choice of model and modifications made to the model in most tasks |
· provided evidence to justify the choice of model and modifications made to the model in some tasks |
· provided evidence to justify the choice of model in some tasks |
· provided little or no justification for the choice of model. |
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· clearly acknowledged the use of all technology. |
· clearly acknowledged the use of technology in most tasks. |
· clearly acknowledged the use of technology in some tasks. |
· acknowledged the use of technology in a few situations. |
· rarely acknowledged the use of technology. |
The Problem:
A manufacturer of floor tiles has hired you as a consultant to design a new square floor tile. The company is prepared to provide for two colours in the tile (light and dark) but lacks the necessary equipment to have more than three simple regions for the colours. You have learned that the manufacturer likes symmetry and dislikes linear patterns.
The manufacturer is particularly concerned about the relative area covered by each colour as this has a marked effect upon the production costs. The diagram below gives an example of an acceptable tile design and corresponding floor pattern given to you by the manufacturer.
One way to model the curves enclosing one of the coloured regions is to approximate them to mathematical functions. If you consider the tile to be one unit square, then the acceptable tile design (such as the one shown above) can be generated by selecting a suitable function in conjunction with its inverse function.
TASK 1: (C&J, K&P)
(a) Determine the inverse function of
.
(b) Describe why the function,
, would be a possible ‘suitable function’ to generate the acceptable tile design above.
(c) Using this particular model, determine the percentage of each coloured area in the tile using
i) a suitable facility on your graphics calculator (print your screen)
ii) a calculus approach (showing full working).
TASK 2: (C&J, K&P)
The function
and its inverse can also be used to produce a similar 2-colour design on a tile which is 5 units square.
(a) Determine the inverse function of
.
(b) Use your graphics calculator or computer application to draw a representation of this tile (include a printout of the result) (You can used the technology to provide the 2-colour shading, or alternatively, simply colour the printout by hand.)
(c) Calculate the approximate percentage of each colour required using the trapezoidal rule with
(i) 5 strips
(ii) 10 strips
(d) Discuss why you are unable (at this stage of your education) to confirm the accuracy of your answers to part (d) above by using a calculus approach.
(e) Use your graphics calculator to confirm the accuracy of your answers to part (d) above. Comment on your findings.
TASK 3: (C&J, M&P)
Although the manufacturer disapproves of linear patterns, he has accepted the following tile design shown below. Here, one of the boundary curves is linear and the other is non-linear; however, the manufacturer insists that the enclosed coloured region alternates either side of the linear function.
The manufacturer demands that the tile be of any suitable size except that of 1 unit square. Present suitable models for the two enclosing functions, and determine the proportion of each colour in your tile. Use a graphics calculator or computer graphing application to generate an accurate picture of this particular tile. Demonstrate that your tile will tessellate successfully.
TASK 4: (C&J, M&P)
An alternate design could be made using parts of two circles as described below.
Find suitable circle equations which would enable the tile to be comprised of two colours in exactly the same proportion. Justify your answer. Use a graphics calculator or computer graphing application to generate an accurate picture of this particular tile. Demonstrate that your tile will tessellate successfully.
(NOTE: Circles centred at the origin with radius r have the general form
.)
TASK 5: (C&J, M&P)
In task 1, you investigated the coloured region enclosed by the function
and its inverse function when used on a unit square tile.
In this task, you are required to investigate the area of the coloured region enclosed by the general function,
, and its inverse function when used on a unit square tile, where n is non-negative.
Present a clear outline to the manufacturer showing how he can easily control the ratio of one colour to the other by varying the value of n in the function used.
Your outline must include the following:
· a general rule (in the form of a fraction developed using a calculus method) which could easily be used to control the colour ratio as desired by the manufacturer.
· a spreadsheet showing the colour ratio for
in increments of 0.25 using this general rule
· a discussion on any abnormalities in this general rule, as evident from the spreadsheet
· a comment on the situation where n is negative.