Activity: The Big Bang Theory  Fractals and complex numbers
AOs 
Indicators 
Outcomes 
Snapshot 
Learning experiences
Cross curricular 
Assessment 
Spotlight 
Links 
Connections
Purpose
To explore complex numbers in the context of the Julia and Mandelbrot Sets.
Achievement objectives
 M 89 Manipulate complex numbers and present them graphically
 M 810 Identify discontinuities and limits of functions
 M 82 Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions
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Indicators
 Uses and manipulates surds and other irrational numbers.
 Interprets and uses graphical representations of complex numbers, using polar and rectangular form on an Argand diagram.
 Uses and manipulates complex numbers, making links with their graphical representation.
 Links features of graphs with the limiting behaviour of functions.
 Uses limiting features of functions to sketch graphs.
 Demonstrates understanding of inverse and/or reciprocal functions, including those of:
 polynomial
 Finds limits algebraically, graphically, and numerically by considering behaviour as:
 x approaches a specific value from above and below
 x tends towards +∞ or ∞
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Specific learning outcomes
Students will be able to:
 Solve problems that require an extended chain of reasoning.
 Understand and iterate functions.
 Manipulate real and complex numbers.
 Use Argand diagrams.
 Simplify sums, differences, products and quotients of surds.
 Simplify sums, differences, products and quotients of complex numbers expressed in rectangular form.
 Solve quadratic equations.
 Sketch the graph of a parabola
 Solve problems involving points of intersection between lines and conics.
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Diagnostic snapshot(s)
Diagnostic snapshot
A fractal is a fragmented geometric shape that can be split into parts, each of which is a reducedsize copy of the whole. Roots of the idea of fractals go back to the 17th Century, however, the term fractal was coined by Benoît Mandelbrot in 1975. A mathematical fractal is based on an equation that undergoes iteration.
Guiding questions:
 Can you draw the function f(x) = x^{2} – 3.25?
 Can you find f(1), f(5), f(3) and f(3 + 2i)?
 What does i represent?
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Planned learning experiences
Starter
Watch the first seven minutes of The Big Bang Theory, Season 1, Episode 2.
(This link is to a YouTube clip. As the link may not be maintained, it may be more appropriate to buy or rent the DVD to show the class.)
Use Google images to find some more Mandelbrot and Julia Fractals.
And now for the maths:
 Take: f(x) = x^{2} – 0.75
 Find f(1). This is a_{1.}
 Find f(a_{1}). This is a_{2}.
 Find f(a_{2}). This is a_{3}.
 Continue this process recording your answers in the chart over:
Chart
 Describe this sequence of numbers.
 Is the sequence approaching any particular value?
 What has this got to do with fractals?
Learning activity
Starting a fractal:
 Draw the curve y = x^{2} – 0.75 on a set of axes between 2 and 2 in red.
 Draw the line y = x in blue.
 Mark your a_{ } to a_{9} values along the xaxis.
 Join your a_{ } value to the red curve with a vertical line, from this point go horizontally across until you meet the blue line. Come down from this point to the xaxis. This should be your a_{1} value.
 Join the a_{1} value to the red curve with a vertical line, and continue the pattern. This begins your fractal. The more iterations you do, the more ‘fractal like’ it becomes.
Experiment 1
 With f(x) = x^{2} – 0.75 try different starting values of a_{ }. For example, try 0.5, 0.8, 1.2, 1.8 and 2.
 What do you notice? Talk about your theories with people in your class. Can you generalise your findings?
Experiment 2
 Try a different base function. It can be from the same family of graphs, or a different one. What do you notice?
Experiment 3
 Try a sample function within a complex number framework: f(z) = z^{2} – 0.75, your a_{ } value is i.
 What happens here? How many iterations do you have to do to decide the outcome?
 Try other imaginary numbers (for example, 0.5i) as your a_{0. }Plot this on an Argand diagram.
 Try some complex numbers (for example, 1/2 + 3i/4) as your a_{ }.
Possible adaptations to the activity
Experiment 4
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Cross curricular links
Extension/enrichment ideas
 Investigate multiple base functions.
 Investigate fractal types.
 Investigate the history of Benoit Mandelbrot or another modern Mathematician.
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Planned assessment*
This teaching and learning activity could lead towards assessment in the following achievement standard:
 AS91577 Mathematics and statistics 3.5 Apply the algebra of complex numbers in solving problems  5 credits; External
* Level 3 achievement standards registered and published in December 2012.
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Spotlight on
Pedagogy
 Creating a supportive learning environment:
 Accepting all student responses as part of the learning process.
 Expecting all students to be able to do mathematics and statistics with understanding.
Key competencies
 Thinking:
 Students select appropriate methods and strategies when solving problems.
 Students make deductions, they justify and verify, interpret and synthesise and they create models.
 Students hypothesise, investigate, analyse and evaluate.
 Using language, symbols and texts:
 Students use symbols and diagrams to solve problems.
 Relating to others:
 Students listen to others, they accept and value different viewpoints.
 Students work in groups, they debate solutions, negotiate meaning and communicate thinking.
 Managing self:
 Students develop skills of independent learning.
 Students are prepared to take risks, make decisions, and persevere.
 Participating and contributing:
 Students contribute to a culture of inquiry and learning. They share strategies and thinking and they empower and enable others.
Values
Students will be encouraged to value:
 Innovation, inquiry and curiosity by thinking critically, creatively, and reflectively
 Community and participation for the common good
Māori/Pasifika

Ka Hikitia

Pasifika Education Plan
 It may be appropriate to investigate whether there are other Mathematicians in different cultures who have investigated fractals since the 17^{th} century.
Planning for content and language learning

ESOL Online
 Reinforce vocabulary such as function, iteration etc.
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Links
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Connections
 Julia and Mandelbrot Sets. David E. Joyce (August, 1994).
Last updated July 30, 2015
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