**Paper 3 investigations**

**Introduction**

Below are a selection of the Paper 3 investigations I’ve made over the years. Many of these bridge between Paper 3 practice (exposure to novel or new mathematical ideas) and the Exploration coursework. All of these could be easily adapted to make some very interesting coursework submissions.

If you are a teacher and find these useful, then please also visit my new site: intermathematics.com for over 3000+ pages of content for teaching IB mathematics including worksheets, mock exams, investigations, enrichment tasks, technology guides, exploration support and investigations.

**Exploration 1: Rotating curves: **

Students explore the use of parametric and Cartesian equations to rotate a curve around the origin

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**Exploration 2: Who killed Mr. Potato?**

Students explore Newton’s Law of Cooling to predict when a potato was removed from an oven.

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**Exploration 3: Graphically understanding complex roots **

Students explore graphical methods for finding complex roots of quadratics and cubics.

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**Exploration 4: Avoiding a magical barrier **

Students explore a scenario that requires them to solve increasingly difficult optimization problems to find the best way of avoiding a barrier.

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**Exploration 5 : Circle packing density **

Students explore different methods of filling a space with circles to find different circle packing densities.

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**Exploration 6: A sliding ladder investigation**

Students find the general equation of the midpoint of a slipping ladder and calculate the length of the astroid formed.

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**Exploration 7: Exploring the Si(x) function **

Students explore methods for approximating non-integrable functions and conclude by approximating pi squared.

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**Exploration 8: Volume optimization of a cuboid **

Students start with a simple volume optimization problem but extend this to a general case of an m by n rectangular paper folded to make an open box.

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**Exploration 9: Exploring Riemann sums **

Students explore the use of Riemann sums to find upper and lower bounds of functions – finding both an approximation for pi and also for .

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**Exploration 10 : Optimisation of area **

Students start with a simple optimisation problem for a farmer’s field then generalise to regular shapes.

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**Exploration 11: Quadruple Proof **

** **Students explore 4 different ways of proving the same geometrical relationship.

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**Exploration 12: Circumscribed and inscribed polygons **

Students explore different methods for achieving an upper and lower bound for pi using circumscribed and inscribed polygons.

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**Exploration 13: Probability Generating function **

Students explore the link between algebra and probability through generating functions and model the eruption of a volcano.

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If you are a teacher and find these useful, then please also visit my new site: intermathematics.com for over 3000+ pages of content for teaching IB mathematics including worksheets, mock exams, investigations, enrichment tasks, technology guides, exploration support and investigations – thanks!

Andrew