Mathematics B (2008) Sample assessment instrument and student

Mathematics B (2008) Sample assessment instrument and student responses

Extended modelling and problem solving June 2010

2 | Mathematics B (2008) Sample assessment instrument and student responses Extended modelling and problem solving

Purposes of assessment1 The purposes of assessment are to:

promote, assist and improve student learning

inform programs of teaching and learning

provide information for those people — students, parents, teachers — who need to know about the progress and achievements of individual students to help them achieve to the best of their abilities

provide information for the issuing of certificates of achievement

provide information to those people who need to know how well groups of students are achieving (school authorities, the State Minister for Education and Training and the Arts, the Federal Minister for Education).

It is common practice to label assessment as being formative, diagnostic or summative, according to the major purpose of the assessment.

The major purpose of formative assessment is to help students attain higher levels of performance. The major purpose of diagnostic assessment is to determine the nature of students’ learning, and then provide the appropriate feedback or intervention. The major purpose of summative assessment is to indicate the achievement status or standards achieved by students at a particular point in their schooling. It is geared towards reporting and certification.

Syllabus requirements Teachers should ensure that assessment instruments are consistent with the requirements, techniques and conditions of the Mathematics B syllabus and the implementation year 2009.

Assessment instruments2 High-quality assessment instruments3:

have construct validity (the instruments actually assess what they were designed to assess)

have face validity (they appear to assess what you believe they are intended to assess)

give students clear and definite instructions

are written in language suited to the reading capabilities of the students for whom the instruments are intended

are clearly presented through appropriate choice of layout, cues, visual design, format and choice of words

are used under clear, definite and specified conditions that are appropriate for all the students whose achievements are being assessed

have clear criteria for making judgments about achievements (these criteria are shared with students before they are assessed)

are used under conditions that allow optimal participation for all

are inclusive of students’ diverse backgrounds

allow students to demonstrate the breadth and depth of their achievements

only involve the reproduction of gender, socioeconomic, ethnic or other cultural factors if careful consideration has determined that such reproduction is necessary.

1 QSA 2008, P–12 Assessment Policy, p. 2.

2 Assessment instruments are the actual tools used by schools and the QSA to gather information about student achievement, for

example, recorded observation of a game of volleyball, write-up of a field trip to the local water catchment and storage area, a test of number facts, the Senior External Examination in Chinese, the 2006 QCS Test, the 2008 Year 4 English comparable assessment task.

3 QSA 2008, P–12 Assessment Policy, pp. 2–3.

Queensland Studies Authority Revised: June 2010 | 3

Mathematics B (2008)

Sample assessment instrument and student responses

Extended modelling and problem solving

Compiled by the Queensland Studies Authority

June 2010

The QSA acknowledges the contribution of Home Hill State High School in the preparation of this document.

About this assessment instrument

The purpose of this document is to inform assessment practices of teachers in schools. For this reason, the assessment instrument is not presented in a way that would allow its immediate application in a school context. In particular, the assessment technique is presented in isolation from other information relevant to the implementation of the assessment. For further information about those aspects of the assessment not explained in this document, please refer to the assessment section of the syllabus.

This sample provides opportunities for students to demonstrate:

recall, access, selection and application of mathematical definitions, rules and procedures

use of number and spatial sense

selection and use of mathematical technology

application of problem-solving strategies and procedures to identify, interpret, clarify and analyse problems

identification of assumptions (and associated effects), parameters and/or variables during problem solving

representation of situations by using data to synthesise mathematical models and generate data from mathematical models

interpretation and use of mathematical terminology, symbols and conventions

organisation and presentation of information for different purposes and audiences, in a variety of modes, e.g. written, symbolic, pictorial and graphical

development of coherent, concise and logical sequences in a response expressed in everyday language, mathematical language or a combination of both, as required, to justify conclusions, solutions or propositions.

This sample assessment instrument is intended to be a guide to help teachers plan and develop assessment instruments for individual school settings.


Assessment instrument

Task: Extended modelling and problem solving

Investigate the relationship between quadratic functions and cubic functions in the living and the non-living structures of our three-dimensional world by examining the relationship between surface area and volume. In particular, you should examine how a change in the linear dimensions of a structure can produce markedly different degrees of change in the surface area and volume of the object.

Such a study has wide implications. It helps to explain, for example, why animals are shaped as they are; how to design more effective packaging; and why construction methods that work well in a metre-high scale model will not work in a 100 m high office tower.

Part A: Preliminary investigation: the cube

1. Write down the formulae for the surface area “S” and the volume “V” of a cube of edge length

“x.” 2. Plot the following graphs:

a S vs x and V vs x on the same axes

b V vs S (Use a computer program such as ‘Autograph’ )

3. Use your graphs to answer the following questions.

For what value(s) of x is:

(i) S > V (ii) S = V (iii) S < V (ignore the difference in units)

4. Use the slope along the curves to find the values of x for which:

(i) S is increasing more rapidly than V.

(ii) V is increasing more rapidly than S.

Show this clearly on a set of graphs.

5. Describe clearly in written English how the relationship between surface area and volume

changes as edge length goes from very small values to very large values.

6. Illustrate your answer to question 5 by plotting the graph of V

S against x.


Part B: Modelling animal structures

The shapes of many animals can be approximated by a series of cylinders, prisms, spheres and cones. Reasonable approximations to their surface areas and volumes can thus be calculated. Tails and ears should be neglected in these calculations unless they are significant in size.

The following are the measurements of some animals:

Animal Length x Surface Area S Volume V

Polar bear 2.40 m 9.585 m2 1.70200 m3

Snowy owl 0.60m 0.266 m2 0.00661 m3

Musk ox 1.90 m 7.430 m2 1.34600 m3

Badger 0.80m 0.854 m2 0.05210 m3

Wolf 1.25 m 2.360 m2 0.98000 m3

1. You have been randomly assigned an ancient Australian animal. (Each student has a different

animal.) You have been provided with a sketch of the animal as well as the length of the animal. Use these to determine a suitable scale for the sketch.

2. Use the scale to calculate all the dimensions of the animal which are required to determine its

volume and surface area. (Note: Re-draw the animal using basic shapes, e.g. cylinders, prisms etc. Place the dimensions on each shape. Label the shapes A, B, C …..)

3. Calculate a suitable value for both the volume and surface area of the animal.

4. Calculate the ratio of V

S for your animal.

5. Plot the ratios V

S against the length (x) of your animal, plus all the animals in the table above.

6. How does this graph compare with the graph in question 6 in the investigation of the cube

above? Present this as a concise written discussion which clearly shows any similarities and differences.

7. What “problems” can you anticipate for animals with:

(i) a large V

S ratio? (ii) a small

V

S ratio

(Note: Use the internet references given in the assignment introduction. At least two paragraphs of concise relevant insights are required here.)


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In completing the mathematical model above and applying the results, take care to: Communicate clearly, logically and concisely so that your implementation is clearly understood Identify assumptions you have made and there associated effects, parameters and/or variables Investigate and evaluate the validity of your mathematical arguments. State the strengths and limitations of your model and also of the information provided in the table above.

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Part C: Modelling the packaging of manufactured goods

In designing containers for their goods, manufacturers aim to keep costs to a minimum. They therefore want to produce containers that have the smallest surface area for the required volume/capacity. However, to produce containers that are functional, compromises in surface area often need to be made.

Task: Investigate the standard 375mL soft drink can.

As can be seen from a diagram of a drink can, it is not a simple cylinder.

Calculate its actual surface area and capacity as accurately as possible.

Is the capacity of the can exactly 375mL?

What might be the reason for this?

Does this can have the minimum surface area for the required volume?

Develop a mathematical model of a perfectly cylindrical can holding exactly 375mL, with variable heights and diameters. Use the model to find the dimensions of a 375mL can with the smallest surface area.

Using the data gathered in the investigation above, suggest reasons why manufacturers may have chosen the present dimensions for the soft drink can. (Note: Only consider “plain” cylindrical shaped containers for this part of this question.)

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In completing the mathematical model above and applying the results, take care to: communicate clearly, logically and concisely so that the implementation of your model is clearly understood. identify assumptions you have made and their associated effects, parameters and variables investigate and evaluate the validity of your mathematical arguments. state the strengths and limitations of your model .

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Instrument-specific criteria and standards

Schools draw instrument-specific criteria and standards from the syllabus dimensions and exit standards. Schools will make judgments about the match of qualities of student responses with the standards descriptors that are specific to the particular assessment instrument. While all syllabus exit descriptors might not be assessed in a single assessment instrument, across the course of study, opportunities to demonstrate all the syllabus dimensions and standards descriptors must be provided.

The assessment instrument presented in this document provides opportunities for the demonstration of the following criteria:

Knowledge and procedures

Modelling and problem solving

Communication and justification.

This document provides information about how the qualities of student work match the relevant instrument-specific criteria and standards at standards A and C. The standard A and C descriptors are presented below. The complete set of instrument-specific criteria and standards is in the appendix.

Standard A Standard C

The student’s work has the following characteristics:

recall, access, selection of mathematical definitions, rules and procedures in routine and non-routine simple tasks through to routine complex tasks, in life-related and abstract situations


recall, access, selection of mathematical definitions, rules and procedures in routine, simple life-related or abstract situations

application of mathematical definitions, rules and procedures in routine and non-routine simple tasks, through to routine complex tasks, in life-related and abstract situations

application of mathematical definitions, rules and procedures in routine, simple life-related or abstract situations

numerical calculations, spatial sense and algebraic facility in routine and non-routine simple tasks through to routine complex tasks, in life-related and abstract situations

numerical calculations, spatial sense and algebraic facility in routine, simple life-related or abstract situations

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appropriate selection and accurate use of technology.

selection and use of technology.


use of problem-solving strategies to interpret, clarify and analyse problems to develop responses from routine simple tasks through to non-routine complex tasks in life-related and abstract situations


use of problem-solving strategies to interpret, clarify and develop responses to routine, simple problems in life-related or abstract situations

identification of assumptions and their associated effects, parameters and/or variables

use of data to synthesise mathematical models and generation of data from mathematical models in simple through to complex situations

use of mathematical models to represent routine, simple situations and generate data

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investigation and evaluation of the validity of mathematical arguments including the analysis of results in the context of problems; the strengths and limitations of models, both given and developed.

interpretation of results in the context of routine, simple problems routine, simple problems.



appropriate interpretation and use of mathematical terminology, symbols and conventions from simple through to complex and from routine through to non-routine, in life-related and abstract situations


appropriate interpretation and use of mathematical terminology, symbols and conventions in simple routine situations

organisation and presentation of information in a variety of representations

organisation and presentation of information

analysis and translation of information from one representation to another in life-related and abstract situations from simple through to complex and from routine through to non-routine

translation of information from one representation to another in simple routine situations

use of mathematical reasoning to develop coherent, concise and logical sequences within a response from simple through to complex and in life-related and abstract situations using everyday and mathematical language

use of mathematical reasoning to develop sequences within a response in simple routine situations using everyday or mathematical language

coherent, concise and logical justification of procedures, decisions and results

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justification of the reasonableness of results. justification of procedures, decisions or results.

Sample student responses: Standard A


Standard descriptors

Student response A

Application of mathematical definitions and rules in simple routine abstract situations

Appropriate selection and accurate use of technology

Use of mathematical terminology, symbols and conventions in simple non-routine situations

Standard A



Appropriate interpretation and use of mathematical terminology, symbols and conventions in simple non-routine situations

Standard A


Demonstration of spatial sense in non-routine, simple, life-related situation

Standard A


Generation of data from a model in a complex situation

Application of mathematical rules and procedures in a non-routine, simple, life-related situation

Further calculations of the volumes and surface areas of the different sections have been omitted from this document. The final summary of these calculations now follows.

Standard A


Use of mathematical reasoning to develop coherent, concise and logical arguments

Numerical calculations in non-routine, simple, life-related situations

Standard A


Organisation and presentation of information in a variety of representations

Analysis and translation of information from one representation to another in simple, non-routine life-related situations

Interpretation of results in the context of a simple mathematical model

Justification of the reasonableness of results

Standard A


Once again the calculations for the can in part C have been omitted, with only the summary results included

Standard A


Justification of the reasonableness of results

Use of problem-solving strategies to interpret, clarify and analyse problems to develop a response in a simple, non-routine, life-related situation

Numerical calculations in a non-routine, simple life related situation

Standard A


Use of mathematical reasoning to develop coherent, concise and logical sequences within a response in a simple, non-routine life-related situation


Coherent, concise and logical justification of procedures, decisions and results

Standard A


Comment:

Some strengths and limitations are identified in the context of the problems but investigation and evaluation of the validity of arguments lacks mathematical justification

Standard C


Standard descriptors

Student response C

Use of mathematical terminology, symbols and conventions in a simple situation

Standard C


Selection and use of technology

Standard C


Selection and use of technology

Standard C


Standard C


Numerical calculations and spatial sense in a simple life-related situation

Standard C


Use of mathematical models to represent routine simple situations, and generate data

Application of mathematical rules and procedures in a routine, simple, life-related situation

Further calculations of the volumes and surface areas of the different sections have been omitted from this document. The final summary of these calculations now follows.

Standard C


Organisation and presentation of information


Standard C


Translation of information from one representation to another in simple routine situations

Standard C


Justification of decisions or results

Standard C


Use of problem-solving strategies to develop a response in a routine simple life-related situation

Numerical calculations, spatial sense in routine simple life-related situations

Standard C



Instrument-specific criteria and standards


Standard A Standard B Standard C Standard D Standard E






recall, access, selection of mathematical definitions, rules and procedures in routine, simple life-related or abstract situations


use of stated rules and procedures in simple situations


statements of relevant mathematical facts

application of mathematical definitions, rules and procedures in routine and non-routine simple tasks, through to routine complex tasks, in life-related and abstract situations

application of mathematical definitions, rules and procedures in routine or non-routine simple tasks, through to routine complex tasks, in either life-related or abstract situations

application of mathematical definitions, rules and procedures in routine, simple life-related or abstract situations



numerical calculations, spatial sense and algebraic facility in routine, simple life-related or abstract situations

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appropriate selection and accurate use of technology

appropriate selection and accurate use of technology

selection and use of technology use of technology use of technology


use of problem-solving strategies to interpret, clarify and analyse problems to develop responses from routine simple tasks through to non-routine complex tasks in life-related and abstract situations


use of problem-solving strategies to interpret, clarify and analyse problems to develop responses to a non-routine simple task in life-related situations involving exponential functions


use of problem-solving strategies to interpret, clarify and develop responses to routine, simple problems in life-related or abstract situations


evidence of simple problem-solving strategies involving exponential functions


evidence of simple mathematical procedures.

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identification of assumptions and their associated effects, parameters and/or variables

identification of assumptions, parameters and/or variables


use of data to synthesise mathematical models and generation of data from mathematical models in simple through to complex situations

use of data to synthesise mathematical models in simple situations and generation of data from mathematical models in complex situations

use of mathematical models to represent routine, simple situations and generate data

use of given simple mathematical models to generate data


appropriate interpretation and use of mathematical terminology, symbols and conventions from simple through to complex and from routine through to non-routine, in life-related and abstract situations


appropriate interpretation and use of mathematical terminology, symbols and conventions in simple non-routine life-related situations involving exponential functions


appropriate interpretation and use of mathematical terminology, symbols and conventions in simple routine situations


use of mathematical terminology, symbols or conventions in simple or routine situations


use of mathematical terminology, symbols or conventions

organisation and presentation of information in a variety of presentations

organisation and presentation of information in a variety of presentations

organisation and presentation of information

presentation of information

presentation of information

use of mathematical reasoning to develop coherent, concise and logical sequences within a response from simple through to complex and in life-related and abstract situations using everyday and mathematical language

use of mathematical reasoning to develop coherent and logical sequences within a response in simple or complex and in life-related or abstract situations using everyday and/or mathematical language

use of mathematical reasoning to develop sequences within a response in simple routine situations using everyday or mathematical language

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coherent, concise and logical justification of procedures, decisions and results

coherent and logical justification of procedures, decisions and results

justification of procedures, decisions or results

Documents

Mathematics B (2008) Sample assessment instrument and student