#### Mathematical Explorations Ten

**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-metryStage: 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

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

This problem gives an opportunity to try something out, notice a pattern, make a conjecture, and then prove it. There are several approaches to the proof, both algebraic and visual, which can lead to fruitful discussion about different methods of proof.

Proving the statements in both directions (if and only if) will be quite challenging for some students, and they will need to persevere and show resilience. To help them to develop these skills, we have provided two "proof sorter" activities which can be used to scaffold the task and offer the support that students need in order to complete the problem.

Skills and concepts experienced: Manipulating algebraic expressions/formulae. Making and proving conjectures. Mathematical reasoning & proof. Proof Sorting. Inequality/inequalities. Triangle numbers. Generalising. Square numbers. Number theory. Patterned numbers.