Brief overview intermediate level
In the module on data compression using songs as an example, students discover how to reduce the storage space of music. In the past, a long-playing record could play about 30 minutes of music per side. Then CDs were developed, which could play about 80 minutes of music. The development of the mp3 compression process was groundbreaking. An mp3 player with a capacity of one gigabyte could store more than 540 minutes, or 9 hours, of music. The music streaming services went even further, managing to make the entire world of music available to you with very little data volume. How can it be that playing and storing music requires so little storage space or so little data?
In the course of the module, the students learn how music is presented on the computer. They then work out their own criteria for a compression method and apply it themselves. At the end of the day, they can compress a selected song using their own method. The guiding question is always: How much can I compress so that there is still good quality?
Among other things, students will have to test their own hearing to develop criteria for compression. Thus, at the end of the module, different procedures may be available and their results compared. In doing so, technical skills from the areas of calculus and trigonometry will be fostered.
Since the module can be varied in difficulty, classes as young as grade 9 can participate in this CAMMP day. For upper level courses, the level will be adjusted accordingly (see below).
Duration: from 5 hours (including lunch break).
Contents: Function equations, trigonometric functions.
Prior knowledge: Function concept
Participants: Mathematics courses from grade 9
Created by: Jonas Kusch, Lars Schmidt, Kirsten Wohak
Registration: Appointments can be made individually using this form.
Image source: https://pixabay.com/de/photos/sängerin-karaoke-mädchen-frau-84874/
Timetable middle school
Phase | Content | School reference | further math. Contents | media/ materials | Time (min.) |
Entry + Technique introduction | Motivation, introduction to the structure of an audio signal, partials as sine/cosine oscillations, simplification and translation of the problem in math. Model | sine/cosine functions | Modeling cycle, Fourier analysis | Presentation slides | 15 10 |
Elaboration AB1 | Tones math. Modeling with different modifications | Understanding data, linear functions, trigonometric functions, graph-function relationship. | - | AB1-SuS | 30 |
Elaboration AB2 | Creating, recognizing and reproducing triads | Trigonometric functions, graph-function relationship, triads and keys | Fourier analysis | AB2-SuS | 40 |
Elaboration AB3 | Investigation of tone sequences using Fourier transform, create and apply own listening model to compress a song. | Trigonometric functions, triads and keys, understanding data | Fourier analysis, auditory models | AB3-SuS | 40 |
Elaboration AB4 | Optimize compression procedures by deleting similar (and too quiet) sounds | Trigonometric functions, understanding data | Modeling cycle, optimization procedures | AB4-SuS | 20 |
Backup + final discussion | Summarize key steps of the compression process, walk through the modeling cycle, discuss occurrences of conflicts, discuss compression | - | - | Presentation slides | 15 |
Additional material (more for upper level) | |||||
Additional sheet: Image compression | Representation of images as matrices, introduction of Haar basis and orthogonal projection, approximation by thresholding | multidim. Vectors and matrices, linear combinations, scalar products orthogonality, programming | standard normal basis, Haar basis, orthogonal projection, thresholding | Supplementary sheet-SuS | 50 |
Brief overview upper level
The structure of the workshop is the same as that for the younger students as explained above, but Fourier analysis is covered in more detail with the upper level students. Students perform the transformation between time and frequency space themselves using matrix multiplications, and independently apply their auditory model in the frequency space by cutting off the frequencies they cannot hear.
The high school workshop can be used as a motivation or application of matrices as they see a use in their everyday life through the workshop.
Duration: from 5 hours (including lunch break)
Contents: Function equations, trigonometric functions, matrices, vectors and their multiplication
Previous knowledge: Function concepts, vectors, matrices (if possible)
Participants: Mathematics courses from grade 11
Created by: Jonas Kusch, Lars Schmidt, Kirsten Wohak
Registration: Dates can be arranged individually using this form.
Timetable upper school
Phase | Content | School reference | further math. Contents | media/ materials | Time (min.) |
Entry + Technique introduction | Motivation, introduction to the structure of an audio signal, partials as sine/cosine oscillations, simplification and translation of the problem in math. Model | sine/cosine functions | Modeling cycle, Fourier analysis | Presentation slides | 15 10 |
Elaboration AB1 | Tones math. Modeling with different modifications | Understanding data, linear functions, trigonometric functions, graph-function relationship. | - | AB1-SuS | 20 |
Elaboration AB2 | Creating triads, storing the audio signal in matrices using an algorithm, reconstruction and determination of triads. | vectors, matrices, matrix-vector multiplication, trigonometric functions, triads and keys, graph-function relation | for loop, | AB2-SuS | 80 |
Backup | Summary of the procedure, going through the modeling cycle. | - | Fourier transform as base change | Presentation slides | 15 |
Elaboration AB3 | Create and apply own listening model to compress a song | vectors, matrices, matrix-vector multiplication, trigonometric functions, triads and keys, understanding data | if loops, | AB3-SuS | 25 |
Elaboration AB4 | Optimize compression procedures by deleting similar (and too quiet) tones | Trigonometric functions, understanding data | Modeling cycle, optimization procedures | AB4-SuS | 15 |
*If still time here continue with additional sheet and then backup with additional sheet. | |||||
Backup + final discussion | Summarize key steps of the compression procedure, go through the modeling cycle, discuss occurrence of conflicts, discuss compression. | - | - | Presentation slides | 15 |
*Additional material | |||||
Supplementary sheet: Image compression | Representation of images as matrices, introduction of Haar basis and orthogonal projection, approximation by thresholding | multidim. Vectors and matrices, linear combinations, scalar products orthogonality, programming | standard normal basis, Haar basis, orthogonal projection, thresholding | Supplementary sheet-SuS | 50 |
Backup + final discussion | Summarize key steps of the compression procedure for data compression of a lid and an image, walk through the modeling cycle, discuss occurrences of conflicts, discuss compression | - | - | Presentation slides | 25 |