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.

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Exploration 1: Rotating curves:

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

Exploration 2: Who killed Mr. Potato?

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

Exploration 3:  Graphically understanding complex roots

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

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.

Exploration 5 : Circle packing density

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

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.

Exploration 7: Exploring the Si(x) function

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

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.

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 .

Exploration 10 : Optimisation of area

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

Exploration 11: Quadruple Proof

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

Exploration 12: Circumscribed and inscribed polygons

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