My Music - Bowland Maths
Contents
1. It may be appropriate to discuss why measures of rate of change use compound units e g m s mph etc how these units of measurement are constructed and to consider some different examples perhaps ones which do not include a time unit e g pressure density etc BPM is constructed from a counted number of beats and a measured period of time There are other possible units e g beats per hour or per second and the teacher may wish to discuss the reasons why a particular one has become standard Beats per minute has an advantage over the other possibilities because the range of values for music or for heart rate is easier to conceptualise and estimate Listening activity taking the pulse of music The teacher is strongly advised to listen to a selection of music tracks before the lesson to check that they are good examples i e that they are easily able to locate the beat and calculate the BPM themselves It is worth pointing out to students that the rhythms in music vary greatly in their complexity and the beat is not always obvious In some music the beat can be quite difficult to find or even ambiguous This is particularly the case with the introduction section of pop songs and it is often a good idea to wait until a little later in the song where the beat usually becomes more obvious It is also a good idea to start the music playing and wait for all or most students to indicate that they have found the pulse by mov2. How does it sound this time Play the track from the start Can you hear when it speeds up and when it goes back to the original tempo Make sure you can still get your music track back to how it was originally If you haven t already done so make sure you save a copy of the original track Then you will always have a backup if you mess up the music you are working on or can t remember what you have done while editing it Try speeding up the music by some other amounts Does it start to sound more like music of a different genre How fast can you make the music before it starts to sound silly Try slowing down the music by 10 To decrease the tempo you need to put a negative sign in front of the number in the Percent Change box Try slowing down the music by some other amounts Does it start to sound more like music of a different genre How slow can you make the music before it starts to sound bad If you have time experiment with some of the other functions in the Effect menu For example you can also change the pitch of the music 33 My Music Name Student Sheet 7 Feedback Who have you been working with Describe what you did during the project How did the project compare to your usual work in Maths Would you recommend it for other teachers to use with their classes Why What maths did you use during the project that you already knew Do you think you learned any new maths If so what Is there anythi
3. 80 BPM 4 groove mp3 120 BPM 5 groove mp3 90 BPM 6 groove mp3 130 BPM 7 groove mp3 100 BPM Cross curricular There may be an opportunity to work with the Music Department on this project In music lessons students could create their own compositions or arrangements of pieces of music which would be appropriate for use in this project particularly in Part 2 With joint planning it may even be possible to co ordinate schemes of work so that students study genre and tempo from a musical point of view in music lessons at the same time as they are analysing them mathematically 21 My Music Web links Students are generally keen to bring their own music from home However if there is any difficulty sourcing music for use by either students or the teacher there is a wide variety of music legally available online Some sites allow music to be downloaded whereas others allow tracks to be played but not saved or copied A selection of useful sites is suggested below Please note that these websites are correct at the time of writing but may be subject to change The iTunes store http www apple com uk iTunes allows unlimited playing of 30 second excerpts of music tracks and has a huge selection The Amazon website http www amazon co uk includes short excerpts from some of the CDs they sell Mp3 com http www mp3 com free music ArtistServer http www artistserver com ArtistGigs http www artistgigs com and Audio
4. There are various commercial audio file conversion programs available some of which offer free trial downloads Teachers taking part in the pre publication trials of this project recommended Smart Audio Converter and dBpoweramp http www smartaudioconverter com http www dbpoweramp com The Windows and Linux but not currently Mac versions of Audacity are also able to extract a low resolution copy of music directly from CD 24 My Music Name Student Sheet 1 Describing music Describe the music you hear Track 1 Track 2 Compare the two music tracks What was different What was the same Differences Similarities Music can be described using variables Examples of variables Think about the variables above e Which depend on personal opinion e Which of them can be counted measured or calculated 20 My Music Name Student Sheet 2 Music and heart rate We are going to measure our heart rates by counting the beats in seconds then multiplying by Your heart rate before Your heart rate after How does this compare to others in the class Your heart rate before Your heart rate after How does this compare to others in the class Think about the results above Do you think that music can affect people s heart rate e Why do you think so 26 My Music Name Student Sheet 3 Taking the pulse of music Find the BPM beats per minute of eac
5. pairs or threes and teacher led whole class activity In both cases there is a mixture of practical activity and discussion Homework may be set either to groups or to individual students The suggested time allowance is 4 lessons for Part 1 and 2 for Part 2 although with some classes it is possible to shorten this The project may be done in a block of consecutive maths lessons or spread out over a longer period for example one My Music lesson per week This latter may be appropriate if there is limited access to computers or audio equipment My Music would work particularly well as a cross curricular project with the involvement of the music department although it does not require any specific musical skills or knowledge My Music Resources Equipment For the first lesson of Part 1 teachers will need the starter tracks provided some music tracks of their own and a means to play them to the class These will most likely be music tracks in mp3 or CD format plus a computer with speakers For the next lesson students will need to bring in their own music to listen to in groups or pairs Some will have their own portable music devices but for others teachers will need to provide listening equipment most likely by booking a set of laptops or a computer suite Similarly not all students will have their own headphones so some may need to be borrowed from the music department Audio splitters are ideal for the practical group work or i
6. players in relation to the concept of randomness how genuinely random is it Lesson 4 Presentation Completion of project The teacher may feel that the project has been rounded off satisfactorily through the previous lesson s whole class activity and discussions and that further time spent on the Part 1 material is unnecessary In this case they may proceed directly to Part 2 17 My Music When writing up their investigation students will probably need reminding of the appropriate structure and content for their report or presentation The amount of guidance needed will vary depending on the age and investigational experience of the students but in all cases it is a good idea to highlight key mathematics words and concepts they might use e g hypothesis method results argument conclusions data category grouped sample random accuracy error mean median range maximum minimum table chart etc If students are using presentation software such as Microsoft PowerPoint or OpenOffice Impress it is wise to check that students are not becoming engrossed in the superficial decorative aspects of making a presentation to the neglect of the mathematical content 18 My Music Supplementary Teaching Notes for Part 2 The amount of time apportioned to this part of the project is up to the teacher It is possible to give a taster of the topic in one lesson but it is likely that 2 hours or more will be needed to
7. practical context and b to introduce students to the process of planning and carrying out an open ended investigation Part 2 uses a similar context task and materials to focus in more depth on rates of change and measurement The majority of subject content falls under National Curriculum Section 3 3 Statistics The only essential prerequisite for students is a grasp of arithmetic although the project works particularly well if they have some prior experience in working with averages and with basic data collection and representation The main thrust of My Music addresses the nature of data and the concept of variables in terms of the characteristics of individual pieces or genres of music Investigations focus on tempo as a measurable variable The project can work as an introduction to statistical project work including the collection of numerical data working out the BPM of music samples performing basic statistical calculations e g average and range of tempos and interpreting the results forming and testing hypotheses based on a sample of music tracks and making inferences about a population i e comparing genres of music However there are also opportunities to focus on several other areas such as the nature of compound measures e g beats per minute factors and multiples and identifying potential sources of error in data collection and calculations Organisation and pedagogy My Music includes both group work preferably
8. the variables suggested Two music tracks teacher s own of contrasting style Which variables are objective and which opinion based Are they qualitative or quantitative Which are measurable Experiment Music and heart rate Resources Direct students attention to one particular variable tempo Encourage SS 2 them to think about why types of music are faster or slower than others Who listens to it Where is it played Do people dance to it Two tusic wack Discuss whether listening to different types of music might affect people s teacher s own one moods slow and one fast Hypothesis Fast music is exciting and slow music is relaxing How could this hypothesis be tested Stopwatch or clock Suggest an experiment to test the effect of listening to fast and slow music on listeners heart rates if students have not suggested this themselves Play two tracks one very slow and one very fast students take their pulse before and after each Discuss results make appropriate calculations and look for patterns If there are anomolous results discuss what might have caused this What other factors might have affected the results e g expectations Lesson 1 continued overleaf My Music Discussion The pulse of music Resources Use the analogy of heartbeats and taking the pulse to introduce the idea An example of a of having a numerical value for the tempo of music Explain the process tempo
9. My Music a mathematical investigation Carla Finesilver My Music Introduction to My Music Overview Listening to music is frequently cited by young people as one of their main pastimes and the music industry is often of interest to them as a potential career choice Many young people also have strong opinions about different types of music with favourite genres and artists My Music utilises this keen interest in music to provide the opportunity for mathematical investigations by using students own favourite music as raw data They make hypotheses about the similarities and differences between genres of popular music with particular focus on tempo They then analyse their music statistically to find out if their hypotheses are correct Throughout the project there are opportunities to address a variety of mathematical topics and issues it is up to the teacher to decide which of these they wish to pursue with a particular class and whether they wish to use the tasks to put into practice previously taught skills and knowledge or as a springboard from which to teach new material My Music is a highly flexible project in which the mathematical content and practical activities may be moulded to suit students of almost any age or ability Mathematical content The case study is in two parts Part 1 has two main objectives a to illustrate or utilise a variety of statistical concepts terms and procedures embedded within a
10. Street http www audiostreet net have free music downloads submitted by a range of independent artists AmazingTunes http www amazingtunes com has full tracks to download from a variety of unsigned artists some to buy others for free Download com http music download com has music tracks from a variety of genres to download or listen online Napster http Awww napstersongs co uk and eMusic http www emusic com have a free trial period for music downloads Many artists and bands use MySpace http music myspace com to allow people to listen to their music 22 My Music ICT Support Part 1 In the first lesson the teacher will need to play music tracks to the class and in subsequent lessons students will need to listen to music in groups This may be done on the computer or using external audio devices such as CD players personal stereos iPods mp3 playing mobile phones etc Most computers will already have audio playing software installed such as iTunes Windows Media Player RealPlayer etc Headphones will be required at least one set for every two students and if working in groups larger than 2 headphone splitters will also usually be necessary Students will very probably want to bring in their own music to use as a source of data This may be in any convenient form such as CDs mp3 files on a memory stick etc If the teacher is planning to use the Audacity software in Part 2 of the project they
11. The suggested way to calculate BPM of heart rate or music is for students to count the beats in a 15 second interval then multiply by 4 to achieve the number of beats per minute i e BPM This is a method with which they are probably familiar from fitness tests in PE Do not spend too long on the practical part of the heart rate activity it has the potential to take up a large chunk of the lesson and for students to become bored There will also probably be some students who find it very difficult to locate their pulse It may be helpful to have students do the pulse taking in pairs but some may not be able to participate fully The resulting data should be discussed at an appropriate level of detail for the class for example ideas of average and range may be introduced or revised at this point as well as inspecting data for any recognisable patterns Is there a general trend e g slowing down then speeding up There is potential for addressing errors in data collection for example by drawing attention to any outliers and discussing how to decide whether a given data point is an error or just an extreme value and how errors might have occurred Discussion The pulse of music Students who have studied a musical instrument will already be aware of numerical tempo markings Printed sheet music is frequently marked by the composer with a suggested BPM or tempo marking to tell performers how fast it is meant to be played e g D 120
12. album or single music track and then ask others to guess the genre band artist from multiple choice answers 11 Resources SS 5 Representational tools as appropriate e g computers number line Back up data set If appropriate this lesson may be omitted proceeding directly to Part 2 My Music Lesson plans for Part 2 The amount of time apportioned to this project is up to the teacher but 2 one hour lessons are suggested It is the responsibility of the teacher to make sure that no illegal copying of music takes place in the classroom At the end of the project all audio files should be deleted from the computers Revise the concept of tempo and its most common unit of measurement Only necessary if BPM beats per minute some time elapsed since Part 1 Listening activity Changing the tempo Introduce students to the idea that the tempo of music may be altered Computer with digitally using special computer software such as Audacity This may be speakers and done in several ways Audacity or other audio editing programme Play the class the original version of a song and then a noticeably installed faster or slower remixed version of it Discuss the effect of the tempo change and why the producer might have chosen to do this _ Some options Interactive whiteboard if available Play an excerpt of a slow song have students calculate the BPM then ask them to suggest an appropriate BPM fo
13. bring out the real mathematics It may be done immediately after finishing Part 1 or later in the year or if time is particularly short not at all It is ideal for cross curricular work with the co operation of one of the school s music and or ICT teachers who might also be willing to devote some of their lesson time to it It is the responsibility of the teacher to make sure that no illegal copying of music takes place in the classroom As an extra precaution at the end of the project all audio files used should be deleted from the computers Lesson 1 Listening activity Changing the tempo Students are generally familiar with the concept of remixes and cover versions of songs but may have thought little about how the process actually takes place The teacher could first encourage them to analyse general differences between two or more versions then focus as in Part 1 on tempo Note that as Audacity manipulates audio signals digitally the change tempo function simply compacts or stretches the music so that playback happens over a shorter or a greater time span It does not have any effect on the pitch of the music in contrast to altering analogue audio signals such as playing a 33 RPM record at 45 RPM or spooling a tape at a different speed Of course it is possible to achieve the analogue effect of concurrent tempo and pitch change digitally through use of the change speed function It is also possible to change pitch alone
14. but these functions are probably best left out of this project at least with younger students It is strongly recommended that even highly experienced teachers and musicians spend some time playing with Audacity or other audio editing software before using it in a lesson Discussion Describing changes mathematically It is important to emphasise the relationship between the tempo at which a section of music is played the BPM of that section and the length of time it takes to play from start to finish Some students may not necessarily realise that a change in one of these things will bring about an equivalent change in the others Group activity Introduction to audio editing With more advanced classes the teacher may wish to discuss the non linear effect of repeated percentage changes For example increasing the tempo of a track by 10 then 10 again then 10 again is not equivalent to increasing it by 30 This is a potential way to introduce exponential functions and compound interest through an unusual context It can be effective to look at aspects of the visual representation of audio data as provided by Audacity The display shows a dynamic waveform function plotted on two axes with the x axis displaying time as is often the case displayed in minutes seconds and decimal fractions of seconds The y axis displays amplitude i e how loud the music is at each point in time By default the amplitude is displayed in linear uni
15. e on an interactive Audacity installed whiteboard preferably at least 1 Allow students some time to play freely with the software and pora wee familiarise themselves with it Set students the introductory tasks provided on SS 6 Even if not Headphones and setting the tasks this help sheet is useful for reference purposes splitters if available Stress to students that in the following group activity they will need to document carefully any changes they make to their music tracks If appropriate they should express changes in terms of percentage increase or decrease relative to the original track Spare music tracks in a compatible electronic format If appropriate demonstrate the effect of repeated percentage changes Group activity Investigating tempo changes Resources Notes Some investigation options Resources as for e Groups choose a music track to remix They investigate altering the previcustoseon tempo by different amounts and document the process This may mean finding a best tempo i e one which they prefer to the original a best tempo for dancing to the music etc Cileehs Squently that students are Groups investigate by what factor they need to speed up or slow making a note of all down a track before it starts to sound like music of a different genre their tempo E g If you take a slow jams track and speed it up can you make it changes It is very sound like hip hop And the rever
16. f the school does not have these students could share one set of headphones between two with one earpiece each Part 2 may be done immediately after Part 1 at a later point in the school year or if time is very limited omitted entirely It requires one computer for each group of students plus headphones etc Computers need to be loaded with the Audacity software or any similar audio editing package It is very helpful if teachers have access to an interactive whiteboard or data projector for demonstration Teaching pack p4 Overview p6 KS3 National Curriculum links p7 Cross curricular links to the National Curricula for Music and ICT ps Lesson plans providing an outline of the themes and activities for each lesson p14 Supplementary teaching notes providing more detailed notes on pedagogical issues particularly recommended for less experienced teachers and non musicians p21 Music support p23 ICT support p25 Student sheets some or all of which the teacher may choose to use p35 Credits A set of BPM starter tracks is included for the first lesson but whole songs are not provided for reasons of copyright A back up data set of BPM information for 48 songs from 1960 2007 is provided in spreadsheet format The Audacity software is also bundled with the CD ROM version of the project distributed under the GNU General Public License My Music Overview of activities and mathematical content The case study is extreme
17. fer to Student Sheet 6 Teachers may wish to go through the notes and practice tasks themselves before using the materials with their class There is also a Quick Reference Guide and User s Manual available to download from http audacity sourceforge net help documentation Various volunteers have contributed to a set of free Tutorials available to read online at http audacity sourceforge net help tutorials There is also an online forum of Audacity users and a dedicated wiki http audacityteam org forum http audacityteam org wiki Further information about the Audacity project including information on the GNU General Public License GPL under which it is distributed may be found here http audacity sourceforge net about Please note that these URLs are correct at the time of writing Although they have remained constant for some time it is possible that in the future some of the external linked web pages may move However they should be straightforward to find via a search engine 23 My Music Audio file formats Audacity is not able to directly import files in proprietary formats such as wma Microsoft or aac Apple This is because of licensing and patent restrictions Audacity can currently import WAV AIFF AU MP2 MP3 and OGG Vorbis files so those files which cannot be directly imported can often be converted to one of these formats Windows Media Player and iTunes are able to do this for non DRM protected files
18. gs of tracks which are in electronic format The group broadens out their focus to investigate a different variable within their set of music tracks such as a particular time period e g popular songs this year country of origin chart position number of copies sold etc The group investigates further the distribution of BPMs on albums How much variation in tempo does there tend to be in a single album Are there outliers i e one or two slow tracks on a generally fast album and where in the track listing do they occur The group investigates other quantitative variables in their chosen music e g the length time of songs Do songs of different genres tend to be of different lengths How much variation is there in the length of songs by the same artist or band Have average song lengths changed over the years 10 My Music Lesson 3 Representation and analysis Class activity Combining representing and analysing data Gather together results from the different groups to produce a larger data set for discussion display and analysis The aim of this is to ensure conscious engagement with the data and draw out as much meaning as possible from the students statistical calculations Encourage students to reflect critically on the data how it was collected whether it is representative biased etc Ask groups for the conclusions from their investigations These can then form the basis for new hypotheses whic
19. h can be tested statisticually using the new whole class data set Some options Ask students for examples of the BMPs they have calculated Type these into a spreadsheet on interactive whiteboard Means medians and ranges of the whole data set or of subsets may be calculated Students compare the new results with those from their own previous group work and decide whether the new data confirms or denies their conclusions Set up a large visual representation to which students can add their results e g put a large number line on the board wall with students sticking on coloured counters to represent the BPM of each data point This makes it easier to see the shape of the distribution There will probably be cluster effects for some genres while others are more widely scattered If students have not collected enough data or their data is unreliable use the back up data set provided It has the BPM and identifying information for 48 chart topping singles from 1960 2007 Homework One of the activities in Lesson 4 below could instead be set as homework Lesson 4 Presentation Completion of project Some options e Groups each produce a poster computer presentation or verbal report on their investigation Individual students write up a formal report on their investigations Groups prepare a music quiz for other students based on their findings For example they could provide some statistical information about an
20. h of the music tracks you hear Estimate Beats in Actual BPM 15 secs BPM Think about the results above e Did everyone in the class get the same answers e If not why do you think this was Planning our investigation zi My Music Name Student Sheet 4 Investigation Description of our music Our hypothesis or question Tips and reminders BPM stands for beats per minute A good way to work out the BPM of a piece of music is to count the beats in 15 seconds and then multiply it by 4 It can be difficult trying to do too much at once One person in the group should be in charge of timing the 15 seconds and a different person in charge of counting the beats If it is too hard to find the beat in the track you are listening to choose another track To find the mean of a set of numbers add them all together then divide by how many numbers there are in the set To find the median of a set of numbers put them in order of size and the median is the one in the middle If there are two in the middle the median is half way between them To find the mode see which number comes up most often To find the range take the lowest number away from the highest number Think about which of the statistical calculations give you the most useful information 28 My Music SS4 p2 Genre artist band etc Track Estimate Actual BPM BPM Notes and calculations We found out
21. he project Teachers may prefer to invite feedback verbally however the sheet proved useful during trials for obtaining the opinion of those students who were less comfortable speaking in front of their peers 20 My Music Music Support Choosing music tracks For demonstration purposes it is best to use music tracks with a clear consistent beat Teachers will probably want to choose tracks from their own music collection but a few ideas are suggested here with some very broad generalisations on genres It is a good idea for the teacher to listen to the music tracks in advance of the lesson to check that it is reasonably easy to hear the beat and calculate the BPM Quite a lot of pop music seems to be around 120 130 BPM particularly in the period from the 1960s to the 1990s Some possible examples are We Gotta Get Out Of This Place The Loco Motion 60s Spirit In The Sky 2 4 6 8 Motorway 70s Sweet Dreams Temptation 80s Sit Down Song 2 90s Ballads can provide examples of significantly slower tempos but can often be more difficult for some students to identify the beat Ones to try might be All By Myself or My Heart Will Go On 50 60 BPM Faster examples can generally be found in some of the rock subgenres particularly heavy metal A popular fast track with the trial students was Sk8er Boi Avril Lavigne 150 BPM R n B is a term which has changed somewhat in meaning ove
22. imum for each genre can be very effective Conclusions formed by each group from their investigations could now be discussed as a Class If a group thinks that they have proved a hypothesis other groups could be asked if on the basis of their own investigations they agree or disagree It may be possible to re test the stronger hypotheses against the larger whole class data set and or the back up data bank to see if they still seem true In trials students tended to assume that because they had checked their own calculations there could be no error or inaccuracy in their results However with minimal prompting they started to be more critical for example realising that some of the class data came from students who may not have been so careful or that other groups definitions of genres and hence classification of music tracks may not agree with their own Another question is how representative students samples of music tracks are of a particular genre artist band decade etc One way to address this issue is to use one of the music websites which has a large number of searchable tracks e g iTunes introducing or reinforcing the concepts of databases and sampling There are various ways to explore the idea of randomness in sampling Students selection of music tracks has not been random so the effect of this on their results may be discussed Mention the shuffle playlist function on audio playing software and some CD
23. ing a hand in time or tapping the desk before starting to count beats For more able classes it may be appropriate to simply play them the music and allow them to work out the BPM by themselves For less able classes it may be necessary for the teacher to say start and stop to delineate a 15 second interval in the music perhaps with students counting out loud It is likely that there will be some variation in the answers given by students This provides another opportunity for the discussion of levels of accuracy and human error in experimental measurements With some music tracks particularly those with either a complex drum track or none at all the beat is very ambiguous and it is likely that some listeners will count twice as fast as others It can even happen that there are students who count faster or slower by a factor of 3 or 4 This should be fairly clear when comparing students answers as their calculated BPM will be roughly 15 My Music double half triple etc the value of others It is also another opportunity to link to the concept of factors and multiples An interesting feature of this activity is that the perceived speed of a piece of music does not always correspond closely to its actual calculated tempo Due to other factors such as rhythmic patterns texture mood etc it is possible for one track to give the impression of being faster than another when the figures show that it is not Surprising res
24. ints covered in the lesson Decide how to organise groups in friendship groups or by ability musical taste etc Inform students that next lesson they will be working in groups to investigate the tempo of their own choice of music Remind them that they must bring in some music in order to take part in the activity it must be appropriate for use in school and should preferably have a clear constant beat Give examples of some possible hypotheses or questions about music If the school does Which are the fastest and slowest genres The tempo of rock music not have a class set varies more widely than dance music Music has become steadily of headphones faster over the years Chart Number Ones have a higher than average remind students to BPM Musicians make slower music as they get older etc bring their own Organise students into appropriate groups and allow them the remainder of the lesson for planning their investigation If unsure they should h ith rticul toi tigat ricular artistiband Find out which choose either a particular genre to investigate or a particular artist band physical format Homework Think of some hypotheses which could be tested Choose students use for some music to bring in to school for the next lesson music CDs MP3 player etc My Music Lesson 2 Investigation Group activity Investigating tempo Each group decides on the hypothesis or question the
25. ly flexible in terms of structure This teaching learning plan is presented in two parts Part 1 of which is intended to correspond to approximately 4 hours of lesson time but could be more or less depending on what the teacher decides is appropriate for their particular class The time allocation for Part 2 and indeed whether it is included at all depends greatly on the resources available to the school and is equally flexible but an allowance of 2 hours is suggested The main intended mathematical content is shown in bold with additional suggestions of concepts which the teacher may wish to include Part 1 Lesson 1 Introduction Discussion of musical genres Concept of variables Types of variables quantitative or qualitative objective or subjective countable measurable etc The use of variables for categorisation Discussion of tempo Construction of a compound unit of measurement Rates of change Choice of appropriate units of measurement Beats per minute BPM calculations Data collection Mean median range etc Factors and multiples Accuracy and errors in data collection Lesson 2 Investigation Investigating tempo Estimation Forming testable hypotheses or suitable questions to investigate Planning an investigation Working systematically Accurate recording of data collected and BPM calculations Choice of appropriate statistical calculations My Music Lesson 3 Representation and analysis Discus
26. marking e g of counting beats and calculating the BPM i e the number of beats per D 120 minute if students do not suggest this themselves Emphasise that although the vast majority of music has a pulse it can in some cases be difficult to locate and or measure it Highlight the fact that any kind of measure of rate of change will use compound units e g m s mph etc and that these units of measurement are generally constructed from two simple units Listening activity Taking the pulse of music Play one or more examples of drum tracks to the class Draw students SS 3 attention to the pulse by waving a hand in time or tapping the desk and encourage students to do the same Drum and groove Help students learn to calculate the BPM see Supplementary Teaching tracks Notes for more detail For each track allow it to play for a while giving students a chance to find the pulse then say start and stop to indicate the counting period Selection of real Move on to groove tracks play one or more examples to the class music tracks with a until they seem confident at finding the beat and calculating the BPM clear constant Finally calculate the BPM of one or more real music tracks Choose ones peal which have a very obvious beat see Music Support Discuss any variations in the answers given by students and the possible reasons for variation Preparation for investigation Review main po
27. may wish to introduce it at this earlier stage A disadvantage of using Audacity in Part 1 is that tracks must be in a suitable format see below or will need to be converted An advantage is Audacity s waveform display which is a real time visual representation of the volume of sound plotted against time where it is often possible to see the pulse of the music as a series of peaks Part 2 For these activities computers are vital as students will need to use audio editing software The school may well already have suitable software installed on some computers for use in music lessons if not it is suggested that the free open source package Audacity is installed on the network Regarding the music to be edited this is most conveniently done in the form of legally downloaded mp3 tracks There are some websites e g iTunes where full high quality music tracks may be bought and others where promotional excerpts or low resolution versions of music tracks may be listened to or downloaded for free Students may well be familiar with these but a set of suggested websites is provided on the Music Support page There are also various free programs available which may be used to import tracks from audio CDs or other media Audacity The latest version of the Audacity software may be downloaded from http audacity sourceforge net download For a beginner s introduction to the audio editing operations needed for Part 2 of My Music please re
28. ng you can suggest to make the project better 34 My Music Credits Concept and all project materials created by Carla Finesilver Drum and groove tracks created with loops from AussieBeats www aussiebeats com Images in Student Sheet 6 from the Audacity manual Anthony Oetzmann Dominic Mazzoni amp Harvey Lubin 2007 Audio wave image on front cover Yurok Aleksandrovich Image from BigStockPhoto com Development of the project was supported by members of the Creative Mathematics Group based at the Institute of Education University of London Dietmar Kuchemann Dave Pratt Teresa Smart Pat Morton Kwame Glevey lan Jones and Sam Parkin Thanks to the schools which participated in trialling of the project The Moat School Fulham Oaks Park High School Redbridge St Angela s Ursuline Convent School Newham Many thanks also to the individual teachers who agreed to trial early versions of My Music with their classes Sunita Magon Stephen Froggatt Rafe Smallman Aishling Ryan and Sue Soman 35
29. od basis for project work however it is not a prerequisite Even if students do not identify their taste in music as falling into certain genres they will probably be aware that music retailers in both high street and online shops tend to categorise the music they sell However they may not have thought about how this actually works Historically categorisation has been done by requesting artists or bands to declare a genre for their music prior to publication or by record companies prior to distribution or even by the music press However genre identification has become a much more interactive and discursive practice since the growth of internet sites relating to music The teacher may wish to find out something about students preferred genres before starting the project e g by doing a quick survey This may then help organise the students into appropriate groups for their investigational work Although most students enjoy talking about their musical tastes there are those who are unwilling There may also be students who mock others preferences or lack of knowledge of popular music It is up to the teacher to be alert to this and maintain discussion of a positive nature Listening activity Variables to define genres If students are having difficulty with the concept of genres and categorisation it may work better to simply discuss the ways in which two or more music tracks are different and the ways in which they are similar Student
30. ove the sliders to the right Click on the drop down menu on the right and choose CD Audio If it is still not working ask your teacher for help Playing an audio file Use these buttons for playing your music tracks Note that while Pause is on H gt i 4 py A A A A A A some of the other buttons will not work Go to Play Record Pause Stop Goto start end When you start the music playing you will see a vertical line moving from left to right This is the cursor and shows you where you are in the music track There is a time bar along the top of the waveform display showing the length of the track in minutes and seconds and how long it has been playing You can zoom in on the track by left clicking this button 2 and zoom out by right clicking 31 My Music SS6 p2 Selecting a section of music You can select a short section of the music by clicking and dragging The section you have selected will go dark like this Mute Solo Project rate 44100 Selection 0 01 462857 0 03 111474 0 01 64 If this does not work click on the select tool ag then try again You can then do lots of different things to the selected music Changing the tempo Select a section of music Click on Effect at the top of the screen and choose Change Tempo You will get a dialogue box something like this Change Tempo Change Tempo without Changing Pitch by Vaughan Johnson amp Dominic Mazzoni using SoundTouch by Olli Pa
31. plications of ratio and proportion d Accuracy and rounding 3 2 Geometry and measures g Units compound measures and conversions 3 4 Statistics a The handling data cycle b Presentation and analysis of grouped and ungrouped data c Measures of central tendency and spread Curriculum opportunities This project is designed to be in line with all the opportunities suggested in the KS3 National Curriculum guidelines 4 a g My Music Cross curricular links Music Key concepts 1 3 a Engaging with and analysing music developing views and justifying opinions 1 3 b Drawing on experience of a wide range of musical contexts and styles to inform judgements 1 4 a Using existing musical knowledge skills and understanding for new purposes and in new contexts 1 4 b Exploring ways music can be combined with other art forms and other subject disciplines Key processes Opportunities for students to 2 2 a analyse review evaluate and compare pieces of music Range and content 3 f Use of music technologies to create manipulate and refine sounds Cross curricular links ICT Key concepts 1 1 c Applying ICT learning in a range of contexts and in other areas of learning work and life 1 5 b Reviewing and reflecting critically on what they and others produce using ICT Key processes Opportunities for students to 2 2 a use ICT tools and techniques appropriately safely and efficientl
32. r the generations During the trial period of this project it was one of the most popular genres chosen by students to investigate Although not usually very fast rhythms can often be quite complex but it is worth listening out for sections where it is easier to take the pulse e g in Crazy In Love Beyonce 100 BPM To stimulate discussion the teacher may wish to give an example or examples where it is not easy to take the pulse It is easy to find music where the beat is not very clear particularly in the introduction sections Examples of songs where the pulse is clear but not constant are Take Me Out Franz Ferdinand which changes tempo in the middle and Bohemian Rhapsody Queen which contains several sections with different tempos Examples of songs with a degree of ambiguity i e one student could quite reasonably count twice as fast as another are Walking On Sunshine Katrina amp The Waves 110 or 220 BPM and I Feel Fine The Beatles 90 or 180 BPM Starter tracks The teacher may wish to use drum tracks and or groove tracks a drum loop with minimal musical accompaniment as a stepping stone between taking students pulses and calculating the BPM of real music A selection is provided here to download in mp3 format Each lasts approximately 1 minute and is made of a single short loop repeated 30 times The tempo of each is listed below 1 drums mp3 110 BPM 2 drums mp3 140 BPM 3 drums mp3
33. r turning the song into a dance track referring to knowledge of genres from Part 1 Try different tempos to see which sounds best In advance prepare 3 or 4 different versions of a popular song Teacher prepared excerpt one at the original tempo the others either speeded up or selection of music slowed down Play the excerpts and ask students to guess which is__ tracks the original Discussion Describing changes mathematically Ask students how the tempo changes can be described more accurately Optional e g double speed half speed Introduce the concept of percentage spreadsheet increase decrease if students do not suggest this themselves Give software for some examples and practice questions at an appropriate level on demonstrating calculating percentage change repeated percentage change calculations Ensure that the link is clearly made between listening to a song being played at a faster or slower tempo than the original and the equivalent numerical change in the BPM ratio If appropriate discuss the non linear effect of repeated percentage increases i e compound interest Lesson 1 continued overleaf 12 My Music Group activity Introduction to audio editing Resources Notes Organise students into groups each group with a computer SS 6 Some introduction options Give a demonstration of changing the tempo of a music track with Computers with Audacity or other audio editing softwar
34. readings of the pulse for each track checking whether the readings approximately agree and if appropriate taking an average 16 My Music It may be helpful and or interesting at this point to demonstrate the visual representation of sound through the use of audio editing software See notes for Part 2 Lesson 3 Representation and analysis Whole class activity Combining representing and analysing data The aim of this activity is to ensure students conscious and critical engagement with the data and to make sure that they are not simply performing rote calculations The degree to which meaning can be drawn out of the data and statistical calculations will depend on the particular class in question and the teacher will decide on the most appropriate approach It will partly depend on the quantity and reliability of data they have produced in the previous lesson so a back up data set is provided in case students do not collect enough reliable data This consists of the title artist band year of release and BPM of 48 music tracks 1960 2007 The data can be represented in various ways For example the BPMs could be grouped with discussion of appropriate group boundaries and then displayed in a pie chart or histogram For more able classes the data may provide an introduction to box and whisker diagrams however if the class is not yet ready for quartiles layered whisker diagrams showing just the minimum median and max
35. rviainen Percent Change 31 0 Beats per Minute BPM from 100 to 131 Length seconds from Preview Cancel o ox If you type 10 for example into the Percent Change box the tempo of the music will speed up by 10 If you type in 10 the tempo will slow down by 10 If you already know the BPM of your music track you can type it into the first BPM box and then in the second box put the BPM you want the music to be 32 My Music SS6 p3 Saving your work Make a folder on your computer where you can save all your work and audio files from this project When you are editing music tracks it is a good idea to keep a copy of the music in its original form before you start making changes Save this file with an obvious name like mysong original If you have made versions of the track with the tempo increased by 10 and 20 of the original tempo you could save them as something like mysong increase10 and mysong increase20 Practice tasks 1 10 Notes Open one of your audio files in Audacity Choose a music track which is medium speed Try playing it for a bit to check that you can hear it properly Select around 20 seconds of music near to the start of the track Increase the tempo of this section by 10 Play the speeded up section Can you hear the difference Change the music back to how it was before using Edit Undo Repeat the process but this time increase the tempo by 20
36. s are often able to think about musical genre in terms of decades or generations e g comparing their own preferred music to that of their parents The teacher directs students attention to thinking about the characteristics of the variables suggested For example is a given variable objective or opinion based If objective is it measurable If measurable how accurately can it be measured These questions may be addressed individually in pairs groups or as a whole class discussion as the teacher prefers It may be helpful here to play at least one musical example which can then be discussed Does everyone agree that it is fast or slow Does everyone think it has the same mood What is there about the track that it is actually possible for everyone to agree on Some examples of variables in music tempo how fast dynamic range how loud pitch range how high low vocal style instruments involved mood intended audience if how people dance to it etc Students are likely to have encountered similar ideas and activities in music lessons Experiment music and heart rate This is an opportunity for some quick revision work on factors e g by asking for all the different ways to work out the BPM 30s x 2 20s x 3 etc Then discuss the benefits and drawbacks of using small time intervals economy of time and effort versus scaling up of errors or large 14 My Music ones greater accuracy but more time consuming
37. se easy for them to forget all about this when amused by e Each group creates a question or questions for a class music quiz by listening to songs at preparing versions of song excerpts at 3 or 4 different tempos then the wrong tempo asks others in the class to identify which is the correct original speed Some music quiz options Each group changes the tempo of a selection of song excerpts and asks others in the class to say whether they have been speeded up or slowed down and to estimate the factor as a percentage increase decrease Completion of project Review the main ideas covered during Part 2 of the project SS 7 Set further presentation work if desired see the last section for Part 1 13 My Music Supplementary Teaching Notes for Part 1 This section contains detailed notes to support the teacher during the running of the case study It is suggested that teachers with less experience of running open ended investigational work read these notes before starting the project They may also be of use to more experienced teachers as an extra resource to dip into as they include notes on several specific pedagogical issues which arose during the trialling of this case study Lesson 1 Introduction What are musical genres It is likely that students particularly towards the end of KS3 will already be able to distinguish between different genres styles of music and this knowledge provides a go
38. sion of data collected representation Identifying errors in data and analysis Interpretation of results Graphical display of data Thinking critically about the investigational process Representative sampling and randomness Lesson 4 Presentation Presentation of work from previous lessons Explanation of findings Coherent and structured communication of the investigation process results and conclusions Part 2 Lesson 1 Introduction Discussion of the effects of changing tempo Relationship between measures of tempo and time periods Mathematical ways of describing change of tempo Audio editing Using software to make mathematical changes to audio files Lesson 2 Investigation Investigating tempo changes Working systematically and recording actions Estimation Calculations involving time Percentage increase decrease My Music KS3 National Curriculum links Key concepts Opportunities for increasing students Competence 1 1 a c Creativity 1 2 a c understanding of the Applications and implications of mathematics 1 3 a d and Critical understanding 1 4 a b Key processes Opportunities for students to work on their skills in Representing 2 1 a d Analysing 2 2 a p Interpreting and evaluating 2 3 a g and Communicating and reflecting 2 4 a e Range and content 3 1 Numbers and algebra b Rules of arithmetic applied to calculations and manipulations with rational numbers c Ap
39. that Next we are going to investigate 29 My Music Name Student Sheet 5 Combining representing and analysing Table for combining BPM beats per minute data from the whole class We found out that Think about your results Are there any that surprise you If so why e Do you think there might have been any errors in the data If so why e Could the data collection have been organised in a better way If so how My critique 30 My Music Student Sheet 6 Introduction to audio editing with Audacity Opening an audio file If your music is on the computer Click on File at the top of the screen and choose Open from the drop down menu Find the music track you want to use and click on it It should open and look something like this X suitcase_m vY Stereo 44100Hz 32 bit float Mute Solo If your music track does not open it may be in a format that Audacity cannot read If you know how convert it to mp3 ogg or wav If you don t know how ask your teacher for help or try a different music track If your music is on CD Insert the CD and start it playing in whatever audio player you normally use In Audacity click on the record button marked with a red circle It should automatically start recording the music and you should see something like the image above If your music track does not record look at the mixer toolbar joo Pq t Line In M
40. ts but there is also an option of displaying it in deciBels which are logarithmic units When playing a music track in Audacity it is easy to see when the music is louder and softer In 19 My Music music with a heavy beat the pulse is often clearly visible in the form of regular peaks In some versions of the program there is also a Beat Finder function which highlights these peaks automatically Group activity Investigating tempo changes One mistake made by several groups during the trials was to select a short section of music say 20 seconds and calculate the BPM multiplying by 3 change the tempo of that section replay that same section and re calculate the BPM They were very surprised to come up with exactly the same BPM as before not realising that their selected section of music was no longer 20 seconds but shorter or longer depending on whether they had compacted or stretched it Despite being warned that Audacity cannot handle every different kind of audio file some students will probably still try to load files in unsupported commercial formats e g Windows Media This is often because they have their computer prefernces set to hide file suffixes and so cannot tell one kind of file from another It may be worth checking with the ICT department to see if they have previously encountered similar issues in ICT lessons Completion of project There is a sheet provided for students to give written feedback on t
41. ults can provide further opportunities for discussion Preparation for investigation It is suggested that students who identify themselves as having similar taste in music be grouped together so that there is for example an R n B group a Rock group etc Most students are very keen to bring in their own music including those less confident about contributing verbally in class If any student cannot or does not wish to bring in music they can be placed in a group of those who do bring their own or use music provided by the teacher In some of the trials of this project telling the students that they might have to listen to the teacher s music was something of an incentive for them to bring their own Group size will vary depending on levels of access to music playing devices sets of headphones splitters etc In trials groups of two or three were generally found to be best It is possible although more difficult for a student to work individually Groups of over four are not recommended Consider how students will be playing their music It is a good idea to find out in advance if their music is mostly on CDs computer or mobile phones etc If there are constraints on the formats they can use in the classroom warn them of this in advance It may be wise to warn students to consider the suitability of the music they bring in for the project For example there is unfortunately a significant amount of popular music which includes se
42. xist and or homophobic lyrics plus a great deal more which includes potentially offensive language Allowing students to play tracks like this in the classroom could be seen as condoning the language and attitudes expressed Lesson 2 Investigation Group activity Investigating tempo Ideally groups of 2 or 3 students sit around a computer or CD player each student with headphones and a headphone splitter if needed A class set of headphones and probably splitters may be borrowed from the music department but if there are not enough to go round students can share headphones in pairs one ear each Schools with access to language booths music practice rooms etc may wish to make other arrangements Depending on the class different levels of formality in the investigational process may be appropriate For example the teacher should first require students to form hypotheses about the variation in tempo within a single genre or about the differences between two or more genres However it is also possible for the class to go ahead on the practical activity with a looser objective i e See what you can find out about the tempo of your music It is suggested that especially with younger students the practical task be broken down into one student listening to the music and indicating the beat one timing 15 seconds and one counting how many beats are indicated in the time period If time permits it is worthwhile to take three
43. y Curriculum opportunities 4 e Share their views and experiences of ICT considering the range of its uses and its significance to individuals communities and society 4 f Use ICT in other subjects and areas of learning with contexts that are relevant and interesting to them My Music Lesson Plans for Part 1 This section contains outline lesson plans based on 4 one hour lessons For more detailed discussion of pedagogical issues arising during the project please refer to the Supplementary Teaching Notes Less experienced teachers are advised to read through these before starting the project Suggested prompts are in italics SS refers to the student sheets All homework tasks are optional Lesson 1 Introduction Discussion What are musical genres Notes Initiate discussion of types of popular music and the ideas of Make brief notes on classification and genre students musical What are the characteristics of different genres How do you know that a preferences music track is of one genre rather than another Listening activity Variables to define genres Play two contrasting music tracks to the class Students describe the SS 1 tracks and identify differences and similarities between them Introduce the idea of using variables to describe music and make a list with the class of such variables e g tempo dynamic range vocal style instrumentation etc Direct students to think about the characteristics of
44. y wish to SS 4 investigate and chooses a selection of music tracks This might involve focusing on a particular genre band or artist Music playing For each track they first estimate the BPM then calculate it as devices accurately as possible It is suggested they take multiple readings of the same track to improve reliability of results If the readings are close together they could take an average if one result is significantly different H h from the others it may be identified as a probable error eaa phones tand splitters if available Remind students to work out the BPM using the 15 seconds x 4 method and not to listen to the whole length of the track each time Stopwatches if Have a stock of additional music perhaps a few albums as there may available be some groups who have forgotten to bring music of their own Homework Students calculate the BPM of a selection of music tracks chosen by their parents guardians Spare music tracks Extension tasks if needed Resources Notes Some options Second copy of SS 4 for some The group picks a second genre band or artist repeats the groups investigation and makes a comparison between the two sets of data Combine two groups who have investigated different genres e g pair a rock group with an R n B group and direct them to compare and Further information contrast their results on music tracks can often be found on the CD covers or in the metata
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