# Mathematics

## Multicultural Mathematical Activities - Rangoli Patterns

This activity will help students

• Understand reflection in two dimensions.
• Construct the reflections of shapes placed at different angles.

Software Required: An Interactive Geometry package. (Instructions are given for Geometer’s Sketchpad – however they should apply to other packages with only slight modifications. If another program is used, a new template will need to be made. The instructions for this are given in the student help sheet.)

Organisation: Students can work in pairs on this activity.

Overview

Students construct Rangoli patterns. These traditional Indian designs make use of reflection in perpendicular and oblique (45° ) mirror lines. The completed designs can be printed and coloured in – they make attractive designs for greetings cards!

Differentiation

• All students draw lines, using the geometry program’s line tools. They use a series of reflections to produce a Rangoli design. The easiest examples are included in exercise 1.

• In the core activity (exercise 2), students attempt to reproduce given designs.

• The extension task involves ‘starting from scratch’, with students setting up their own patterns and experimenting with designs of different sizes.

Demonstration
First, demonstrate to the class how to produce a Rangoli pattern starting from the ready-made template. Explain how the pattern is formed.

• The basic pattern consists of 5 line segments. These segments are shown in the grey square in the corner of the screen.

• These line segments are reproduced (translated) to form the basis of the main pattern, where they are reflected in a diagonal mirror line to create the top left ‘quarter’ of the design.

• The top left quarter of the design is then reflected vertically and horizontally to complete the pattern.

To change the pattern, drag the line segments (or their end-points) in the grey square. Particularly pleasing patterns are formed when some of the end-points lie on the boundary of the grey square, and end-points within the square coincide; this will cause the completed pattern to ‘join up’.

Once students have had the opportunity to produce some Rangoli patterns of their own, they can start work on the worksheets.

Notes

• The diagram is set up so that points snap to the grid (Graph menu), with measurement units being centimetres (Preferences item from the Display menu.)

• The idea of displaying the ‘unreflected’ lines separately from the main pattern is that this makes it possible to see where these lines actually are; if the construction were set up so that the lines drawn formed the pattern directly, it would be almost impossible to distinguish between the original lines and their various reflections.

• Hiding unnecessary lines (like the mirror lines) and labels (just about everything except the labels on the ‘original’ line segments) will make the final construction much easier to understand.

Teaching Points
It is important to stress some key features of ‘mathematical mirrors’:

• They are ‘two-way’; objects on either side of the mirror line are reflected to the opposite side.

• Images of objects are produced the same distance from the mirror line as the original image, and the direction of reflection is perpendicular to the mirror line.

• Rangoli patterns do not have to be based on 5 line segments; however, 5 segments will produce some very aesthetically pleasing patterns, especially if care is taken to ensure that they ‘join up’ appropriately.

• Most students will enjoy making their own Rangoli patterns and colouring them in to form a symmetrical design. The core task introduces a further challenge, in the form of a game for two players. It is also possible to get students to construct their own version of the Rangoli patterns diagram, as indicated in the extension task. The necessary steps are set out in the student help sheet. Note, however, that this is a fairly demanding construction which is probably best suited to students who are quite confident with the software.