#### Mathematical Explorations Eight

**MatExEight**

Ages 13 to 14

**Exploring Geometry**
Triangles in a Square

Stage: 3

In this problem students explore triangles made in differing sizes of squares formed on square dotty paper. Students explore the smallest and largest areas such a triangle can have as well as the differing triangle areas in between.

Students have the opportunity to find a general expression for the area of a triangle on any grid.
Square Areas

Stage: 3

Students determine the area of the inner square formed with diagonal lines within a larger square of length 1. Students develop skills of visualizing and making Mathematical reasoning & proof. They will have the opportunity to use Pythagoras' theorem to generalize an expression
Cyclic Quadrilaterals

Stage: 3

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. It is well suited for students who are working on circle theorems, or for applying basic understanding of angles in triangles. Initially working on the building blocks gives students the opportunity to then work on harder mathematical challenges than they might otherwise attempt. The problem is structured in a way that makes it ideal for students to work on in small groups.

**Exploring Geometry & 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 generalisations that can be proved with some vector algebra.

Skills and concepts experienced: Vector algebra. Maths Supporting SET. Scalar products. Rotations. Interactivities. Curious. Matrices. Vectors. Vector Notation and Geometry. Squares.

Areas of ParallelogramsStage: 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.

Skills and concepts experienced: Generalising. Vectors. Vector Notation and Geometry. Scalar products. Mathematical reasoning & proof. Area. Parallelograms. Maths Supporting SET. Creating expressions/formulae.

Kite in a SquareStage: 4

Can you make sense of the three methods to work out the area of the kite in the square?

Students often find geometric proofs quite intractable. In this problem, three different ways of proving the same result are presented, jumbled up, so that students can engage with the proofs without having to start from scratch.

Skills and concepts experienced: Inequality/inequalities. Similar triangles. Making and proving conjectures. Cartesian equations of lines. Mathematical reasoning & proof. Pythagoras' theorem. Proof Sorting. Resilient. Number theory. Manipulating algebraic expressions/formulae.