Mathematics C (2008): Sample assessment instrument and

Mathematics C 2008 Sample assessment instrument and student responses

Extended modelling and problem solving October 2009

2 | Mathematics C 2008

______________________________________________________________________________

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 C syllabus and the implementation year 2008.

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.

Sample assessment instrument and student responses | 3

Mathematics C 2008

Sample assessment instrument and indicative responses

Extended modelling and problem solving

Compiled by the Queensland Studies Authority

December 2009

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 instrument provides opportunities for students to:

recall, access, select and apply mathematical definitions, rules and procedures

select and use mathematical technology

apply problem-solving strategies and procedures to identify problems to be solved and interpret, clarify and analyse problems

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

develop and use coherent, concise and logical supporting arguments, expressed in everyday language, mathematical language or a combination of both, when appropriate, to justify procedures, decisions and results.

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


Assessment instrument The student work presented in this sample is in response to assessment items which are subsets or parts of an assessment instrument.

The Mathematics of Music “If music be the food of love, play on.” – William Shakespeare

When students first learn about trigonometry, it is often very difficult to see beyond the basics of “sin”, “cos” and “tan” to get a good appreciation of just how important the topic is in everyday life. Many aspects of our natural world can be defined and explained in terms of periodic functions, including temperature modelling, tidal measurements and the motion of springs and pendulums. Perhaps one of the most important (and enjoyable) applications of periodic functions, however, is sound.

Sound simply consists of travelling waves, involving variations in pressure through a solid, liquid or gas. The most fundamental properties of a basic sound wave are frequency, which affects the pitch of a sound, and amplitude, which affects its loudness. Different instruments (including the voice) produce subtle yet complex changes in the basic wave structure, which give the instruments their characteristic qualities. In this assignment, we will be using trigonometric functions to model basic sounds, and we will investigate how harmonies can be formed by combining multiple soundwaves.

In the following questions, you may need to research how a musical note can be represented by a trigonometric function. (This information can be found on the internet or in your textbook). The questions in Part 1 refer to a data file found on the School Intranet – you will need to locate the numbers corresponding to your individual student number.

PART 1 – THE BASICS OF SOUND

QUESTION 1 (KAPS)

When played, a musical instrument produces a soundwave with a frequency of f vibrations per second and an amplitude of a. Using the unique values of f and a corresponding to your student number in the data file, answer the following questions:

a) Find the angular frequency, ω, of this soundwave assuming a sinusoidal waveform. b) Write the equation of the waveform.

c) Produce a graph of this waveform using a graphing program of your choice (Autograph,

Geogebra, Excel or the Graphics Calculator) over the interval seconds. Attach a printout of your graph.

QUESTION 2 (KAPS)

When two musical instruments are played, a soundwave is made up of two partial waves both having an amplitude of k (from the data table). The period of the fundamental is f1 and that of the overtone is f2 (from the data table).


a) Find the angular frequency, ω, of both partial waveforms.

b) Write the equation of the separate and composite waveforms.

c) Produce a graph of this waveform using a graphing program of your choice (Autograph,

Geogebra, Excel or the Graphics Calculator) over the interval seconds. Attach a printout of your graph.

PART 2 – REAL-LIFE SOUNDS

Audacity is a free, multi-platform, open-source software program designed for sound analysis and editing. It is possible to record or open a sound file using Audacity and view a graphical representation of the waveform. This information can then be used to analyse aspects of the sound such as its pitch and loudness.

A basic demonstration of Audacity will be given in class, using a sample of the 1991 number-one single Rush by Big Audio Dynamite II. This sample is remarkable for the variation in the pitch of lead vocalist Mick Jones, a feature which can be seen graphically in the Audacity program.

For this section of the assignment, you will need to research a list of frequencies of soundwaves and how these relate to the pitch of the sound (in musical form). For example, an A note in the second octave (A2) is equivalent to a frequency of 110 Hz (cycles per second). If the frequency changes, so too does the note.

QUESTION 3 (MAPS)

Download the file mysterynote.wav from the School Intranet. Use the Audacity software to open the file and examine the soundwave. By comparing features of the soundwave with your research, determine which note most closely matches the mysterynote.wav file. Justify your answer mathematically. (You do not need to state the octave of the note, just the letter of the note).

QUESTION 4 (MAPS)

As you would be aware, the mechanical properties of the piano strings and mechanism gradually change over time, and this is why a piano needs to be tuned on a regular basis. Your teacher has recorded a series of notes on his piano at home. They are located on the School Intranet and are labelled with the letter of the note. View the data table to see which note you have been allocated. You should then analyse the waveform produced by this note using Audacity to determine how well-tuned the piano note sounds, relative to the “theoretical” waveform of the same note. Justify your answer mathematically.

QUESTION 5 (MAPS)

Just as harmonies are produced on a musical instrument by playing more than one note simultaneously, they can also be produced mathematically by adding different trigonometric functions together.

a) Using your research and knowledge from previous questions, design a mathematical

equation that would represent a harmony between a C in the 5th octave and an E in the 5th octave. Write the equation and produce a graph of the function in the domain


seconds. Attach a printout of your graph.

b) Using Audacity, load the audio files of the C and E notes from your teacher’s piano. Combine the two notes into a single waveform and zoom in to a similar domain as in part a) above. Attach a screenshot of this waveform.

c) Comment briefly (150 words maximum) on the similarities and differences between the graphs in parts a) and b) above. Explain any major differences between the two graphs.


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 (KAPS)

Modelling and problem solving (MAPS)

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 follow the student responses.

Standard A Standard C

Kn

ow

led

ge

and

pro

ced

ure

s

The student work has the following characteristics:

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

appropriate selection and accurate use of technology.


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

selection and use of technology.

Mo

del

ling

an

d

pro

ble

m s

olv

ing


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.

Co

mm

un

icat

ion

an

d ju

stif

icat

ion

The student’s work has the following characteristics:

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

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

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


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

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

justification of procedures, decisions or results.

Key Differences or additional requirements for demonstrating the standard.

Differences in complexity of task requirements for each standard

Sample student response: Standard A


Standard descriptors Student response A

Application of

mathematical

definitions, rules

and procedures

in a routine and

non-routine

simple tasks

through to

routine complex

tasks, in life =-

related and

abstract

situations

Sample student responses: Standard A


Application of

mathematical

definitions, rules

and procedures

in a routine and

non-routine

simple tasks

through to

routine complex

tasks, in life =-

related and

abstract

situations

Appropriate

selection and

accurate use

of technology



Coherent,

concise and

logical

justification of

procedures,

decisions and

results



Coherent,

concise and

logical

justification of

procedures,

decisions and

results



Coherent,

concise and

logical

justification of

procedures,

decisions and

results



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

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.

Standard C


Note: “[…]” indicates where the text has been abridged.

Standard descriptors Student response C

Application of

mathematical

definitions, rules

and procedures

in a routine and

non-routine

simple tasks

through to

routine complex

tasks, in life-

related and

abstract

situations

Standard C


Standard C


Application of

mathematical

definitions, rules

and procedures

in a routine and

non-routine

simple tasks

through to

routine complex

tasks, in life-

related and

abstract

situations

Standard C


Standard C


Selection and

use of

technology

Standard C


Justification of

procedures,

decisions or

results

Standard C


Use of problem-

solving strategies

to interpret, clarify

and develop

responses to

routine, simple

problems in life-

related situations

Standard C


Use of

mathematical

reasoning to

develop a

sequence within a

response in simple

routine situations

using everyday

language

Instrument-specific criteria and standards

Standard A Standard B Standard C Standard D Standard E K

no

wle

dg

e an

d p

roce

du

res


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

appropriate selection and

accurate use of technology.


application of mathematical

definitions, rules and

procedures in routine complex

tasks, in life-related

trigonometry situations

appropriate selection and

accurate use of technology.


application of

mathematical definitions,

rules and procedures in

routine, simple life-related

or abstract situations

selection and use of

technology.


use of technology.


use of technology.

Mo

del

ling

an

d p

rob

lem

so

lvin

g


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 abstract situations.



strategies to interpret, clarify

and analyse problems to

develop responses to routine

complex tasks in life-related

trigonometry situations.



strategies to interpret,

clarify and develop

responses to routine,

simple problems in life-

related or abstract

situations.


evidence of simple

problem-solving

strategies in the

context of problems.


evidence of simple

mathematical

procedures.


Instrument-specific criteria and standards


Standard A Standard B Standard C Standard D Standard E C

om

mu

nic

atio

n a

nd

ju

stif

icat

ion


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

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

coherent, concise and logical

justification of procedures,

decisions and results


appropriate interpretation and

use of mathematical


conventions in simple non-

routine, in life-related

trigonometry situations

use of mathematical reasoning

to develop coherent and

logical sequences within a

response in complex and in

life-related trigonometry

situations using everyday

and/or mathematical language

coherent and logical


decisions and results


appropriate interpretation

and use of mathematical


conventions in simple

routine situations

use of mathematical

reasoning to develop

sequences within a

response in simple routine

situations using everyday

or mathematical language


decisions or results


use of mathematical

terminology, symbols

or conventions in

routine trigonometry

situations


use of mathematical

terminology, symbols or

conventions

Key Differences or additional requirements for

demonstrating the standard.

Differences in complexity of task requirements for each standard

Documents

Mathematics C (2008): Sample assessment instrument and