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 readymade 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 endpoints) in
the grey square. Particularly pleasing patterns are formed when some of
the endpoints lie on the boundary of the grey square, and endpoints
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 ‘twoway’; 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.
Downloads:
Rangoli Patterns Student Help
Sheet
Rangoli Patterns Exercise 1
Rangoli Patterns Exercise 2
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