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# Activity: Fractals

## Purpose

This teaching and learning activity explores fractals and sequences, and links to the internal assessment resource Mathematics and statistics 2.3D fractals.

## Achievement objectives

• M7-3 Use arithmetic and geometric sequences and series

## Indicators

• Identifies and uses representations such as geometric patterns, coordinates, lists of terms, recursive formulae, general formulae, and sigma notation.
• Solves problems that can be modelled by arithmetic and geometric sequences and series.

## Specific learning outcomes

Students will be able to:

• identify and use features of fractals such as length, area, number of items, volume, and height
• create fractals, understanding the sequential nature of their development
• solve sequences and series problems involving fractals.

## Planned learning experiences

### Familiarisation with fractals

Students will need to know how to draw the fractals before attempting the two activities: the Koch curve and the Sierpinski triangle. The five PowerPoints below are for this purpose, and other resources can be used.

### Activities

There are two activities involving two famous fractals:

The aim of these activities is to familiarise students with the content and expectations of the investigation required by the assessment task Mathematics and statistics 2.3D: Fractals to meet the achievement standard AS91258 Mathematics and statistics 2.3 - Apply sequences and series in solving problems.

An important part of the activities is for students to explore the context of fractals so that they are familiar with the issues that need considering:

• The iterative method used to produce the next stage in a fractal.
• How sequences can be used to find out various features of the resulting fractals.
• Where summing the sequences to form a series, makes sense in the context of fractals.
• Discover what happens to the sequences as the number of iterations increases and see if a sum to infinity makes sense.
• Selecting a suitable limit and solving to find the term in a sequence.

#### Clarification on the use of the term “features” in this context

Features are concepts such as length, area, number of items, volume, height.

Example:

• One feature is area, such as the area of one triangle being added (this would generate a sequence) or the total area (possibly a series) of the fractal for each iteration. A student may choose to investigate more than one sequence using area.
• Another feature is length, such as perimeter of one triangle, the length of one side, total perimeter of the fractal: each has its own sequence.

Note: Students need to investigate at least two features for investigations.

• Art

## Planned assessment

This teaching and learning activity is designed to support the internal assessment activity Mathematics and statistics 2.3D: Fractals:

• AS91258 Mathematics and statistics 2.3 Apply sequences and series in solving problems - 2 credits; Internal

## Spotlight on

### Pedagogy

• Providing sufficient opportunities to learn, by:
• using an engaging starting point for learning
• making learning intentions clear
• modelling and making explicit, desired practice
• checking for student understanding before proceeding
• allowing sufficient time to solve tasks
• providing activities with differentiated entry and exit points, including extension and enrichment.
• e-Learning and pedagogy:
• Use technology to explore concepts.
• Share learning via technology.
• Use tools such as calculators, spreadsheets, and software packages.

### Key competencies

• Thinking:
• Students select appropriate methods and strategies when solving problems.
• Students make deductions, they justify and verify, interpret and synthesis, and they create models.
• Students use mathematics to model real life and hypothetical situations; they make conjectures, challenge assumptions and thinking, and they engage in sense making.
• Students ask questions, want to know ‘why’, make connections, and discern if answers are reasonable.
• Using language, symbols, and texts:
• Students use symbols and diagrams to solve problems.
• Students use ICT appropriately. They capture their thought processes, recording, and communicating mathematical ideas.
• Students interpret word problems and visual representations.
• Managing self:
• Students set goals, rise to new challenges, ask for help if needed, and take ownership of their learning.
• Students plan and manage time effectively.

### Values

• Students will be encouraged to value:
• excellence, by aiming high and by persevering in the face of difficulties
• innovation, inquiry, and curiosity, by thinking critically, creatively, and reflectively
• ecological sustainability, which includes care for the environment.