#### MathematicalExplorationsTen

MatExTen
Ages 15 to 16

Exploring Vectors
A brief introduction to Vectors is included for those new to this topic. Vector Journeys
Stage: 4
This problem offers a simple context for exploring vectors that leads to some interesting generalizations that can be proved with some vector algebra. Areas of Parallelograms
Stage: 4
The problem looks as though it is about vectors and area formulae. It diverts via alternative methods for finding area, into observing, then systematically hunting down a numerical relationship, proving this algebraically and geometrically, and testing where the proofs are valid. It also relates to determinants of matrices which students may encounter in later years. Vector Walk
Stage: 4 and 5
This problem encourages students to think about vectors as representing a movement from one point to another. The need for coordinate representation of points will emerge automatically and the problem naturally requires an interplay between geometry and algebra.

Further Mathematical Study

Hexy-metry
Stage: 4

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

This problem requires the solver to reason geometrically and make use of symmetry. By re-presenting the information in a different way, for example by adding additional lines (a useful technique in geometrical problems) more structure can be revealed. It is an interesting idea that adding something, and therefore apparently making it more complex, can sometimes make a problem more accessible. Then of course there is an opportunity to use the cosine rule in a non-standard context.

Skills and concepts experienced: Probability. Cosine rule. Tree diagrams. Radius (radii) & diameters. Maximise/minimise/optimise. Resourceful. Isosceles triangles. Curious. Pythagoras' theorem.

Vector Walk
Stage: 4 and 5

This problem encourages students to think about vectors as representing a movement from one point to another. The need for coordinate representation of points will emerge automatically and the problem naturally requires an interplay between geometry and algebra.

Skills and concepts experienced: Vector Notation and Geometry. Mathematical modelling. Vectors. Maths Supporting SET. Physics. Scalar products. Biology. Matrices. Real world. Investigations.

Iff
Stage: 4 and 5