**‘Create beautiful polygons and stars using online micro-world and discover for yourself rules for exterior angles.’**

Polygons and stars are visually appealing because of their symmetry. But what is it that lies at the heart of these images? They have a regular mathematical structure, but can you discover it? In this investigation you will use your computer to create some pretty pictures and find the rules behind the angles within the shapes.

**Watch this screencast to get a quick overview of the activity**

**Resources**

- Student activity Polygons & Stars
- Students will need this virtual manipulative Online interactive turtle
- The teacher may wish to read these teacher notes Polygons and stars

**Description**

- Start with pencil, paper, ruler and protractor to produce a figure.
- Use computer with internet access to construct figure with online Logo the interactive Turtle.
- Can you produce a regular polygon?
- Can you produce a regular star?
- When does it not work?
- How is the command linked to the number of sides of the polygon or star?
- Write up your findings in a report including diagrams and formulae used.

# Modelling music

*‘Discover the underlying mathematics behind music’*

Do you like music? If you play a musical instrument you’ll know that some combinations of notes sound great, and others sound dreadful. In this investigation you are going to discover that it is mathematics that lies behind these notes, chords and harmonies. Tuning forks are used to tune instruments. Mathematically they are interesting since they produce pure tone sound – there is no echo like with a piano or guitar body, for example. You are going to listen to some different tuning fork sounds and look at the visual representation of these sounds. By reproducing the graphs of these sounds for yourself you should understand how they are related to the musical scale.

**Resources & Description**

- For this activity you will need some graphing software
- Use graphing software to explore the effect of changing a and b in the equation y = asinbx
- Here is the main part of the activity The Mathematics of Sound to be attempted after watching video instructions
- Check your understanding: try this QUIZ to see if you can apply what you have learned.

**Amplitude & Frequency**

Learn how the value of a affects the amplitude of the sine curve y = asinx

Which of the following graphs have an amplitude of 2 [Show][Hide]

y=sin2x

y=0.5sinx

y=2sinx

y=sin(0.5x)

Explanation:

What is the amplitude of the graph y = -3sin 2x ? [Show][Hide]

2

1/3

-3

3

1/2

Explanation:

Learn how the value of b affects the period of the sine curve y = sinbx

Which of the following graphs has a period of 360°? [Show][Hide]

y=360sinx

y=sin360x

y=sinx

y=sin(x/360)

Explanation:

What is the period of the graph of y=sin2x? [Show][Hide]

2

360°

720°

180°

Explanation:

Which of the following graphs has a period of 30°? [Show][Hide]

y=sin30x

y=sin12x

y=sin(x/12)

y=sin(x/30)

Explanation:

Which of the following graphs has a period of 1°? [Show][Hide]

y=sinx

y=sin(x/360)

y=360sinx

y=sin360x

Explanation:

Which of the following graphs has a period of 0.1°? [Show][Hide]

y=sin3600x

y=0.1sinx

y=sin0.1x

y=sin10x

All answers: [Show] [Hide]

The following applet should help you understand the dynamics of sound Sound Wave

The following short video shows a recording of a pitching fork (pitch A) from http://www.onlinetuningfork.com/and then a visual representation of the sound recorded with Audacity, a sound recording piece of software.

The following video gives an introduction to the activity. Watch it to see the shape of the graph of pitch A4

Video Instructions

A visual representation of the A4 pitch

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