Activity

The Big Bang Theory

Purpose

To explore complex numbers in the context of the Julia and Madelbrot Sets.

Achievement objectives

M 8-9 Manipulate complex numbers and present them graphically

M 8-10 Identify discontinuities and limits of functions

M 8-2 Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions

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:

o  polynomial

·  Finds limits algebraically, graphically, and numerically by considering behaviour as:

o  x approaches a specific value from above and below

o  x tends towards +∞ or -∞

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 including i.

·  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

Diagnostic snapshot(s)

A fractal is a fragmented geometric shape that can be split into parts, each of which is a reduced-size 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) = x2 – 3.25?

·  Can you find f(1), f(5), f(-3) and f(3 + 2i)?

·  What is i?

Planned learning experiences

Starter

http://www.youtube.com/watch?v=ay8OMOsf6AQ Madelbrot talks at TED

http://www.youtube.com/watch?v=G_GBwuYuOOs Zooming in on a Madelbrot Set

Watch the first 7 minutes of The Big Bang Theory, Season 1, Episode 2.

Use google images to find some more Mandlebrot and Julia Fractals.

And now for the Maths:

·  Take: f(x) = x2 – 0.75

·  Find f(1). This is a1.

·  Find f(a1). This is a2.

·  Find f(a2). This is a3.

·  Continue this process recording your answers in the chart over:

a0 / 1
a1 / 0.25
a2
a3
a4
a5
a6
a7
a8
a9

·  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:

1.  Draw the curve y = x2 – 0.75 on a set of axes between -2 and 2 in red.

2.  Draw the line y = x in blue.

3.  Mark your a0 – a9 values along the x axis.

4.  Join your a0 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 x axis. This should be your a1 value.

5.  Join the a1 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) = x2 – 0.75 try different values of a0. 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) = z2 – 0.75, your a0 value is i.

·  What happens here? How many iterations do you have to do to decide the outcome?

·  Try other imaginary numbers (e.g. 0.5i) as your a0. Plot this on an Argand diagram.

·  Try some complex numbers (e.g. 1/2 + 3i/4) as your a0.

Possible adaptations to the activity:

Experiment 4:

·  Investigate plotting Julia and Mandelbrot sets at: http://aleph0.clarku.edu/~djoyce/julia/explorer.html

·  What is the difference between a Julia set and a Mandelbrot Set? Describe this.

Cross curricular links

·  Physics

Extension/enrichment ideas

·  Investigate multiple base functions.

·  Investigate fractal types.

·  Investigate the history of Benoit Mandelbrot or another modern Mathematician.

Planned assessment

This teaching and learning activity could lead towards assessment in the following achievement standard:

·  3.5 Apply algebraic methods in solving problems 5 credits External

Spotlight on

Pedagogy

·  create a supportive learning environment where discussion and analysis is encouraged

·  encourage reflective thought and action

·  enhance the relevance of new learning

·  facilitate shared learning

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 hypothesise, investigate, analyse and evaluate.

·  Students design investigations, explore and use patterns and relationships in data and they predict and envision outcomes.

·  Students deal with uncertainty and variation, they seek patterns and generalisations.

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.

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

·  Compass for Pasifika success

·  It may be appropriate to investigate whether there are other Mathematicians in different cultures who have investigated fractals since the 17th century.

Planning for content and language learning

·  ESOL Online

·  Reinforce vocabulary such as function, iteration etc.

Links

·  http://www.youtube.com/watch?v=ay8OMOsf6AQ

·  http://www.youtube.com/watch?v=G_GBwuYuOOs

·  http://aleph0.clarku.edu/~djoyce/julia/explorer.html

·  http://www.clarku.edu/~djoyce/complex/

Connections

Julia and Mandelbrot SetsDavid E. Joyce August, 1994.

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