Students use a range of physical and digital technologies to model and explore a practical geometric problem. They use dynamic modelling and graphing techniques to model and solve two variations of the problem, then use reflections and geometric reasoning to justify their findings.

This sequence is designed to develop conceptual understandings of geometric proofs and consolidate understandings of Pythagoras’ theorem and similar triangles. It is for students who:

- have been introduced to Pythagoras’ theorem.
- are able to complete routine calculations to find the length of the hypotenuse or of one of the short sides.
- can identify and work with similar triangles

### Lesson 1: How Far?

Students participate in an investigation to find the length of a path that touches three sides of a rectangle, starting and finishing at the same point on the fourth side. They model the problem and gather data on possible solutions.

### Lesson 2: One Corner

In the previous lesson students formed the hypothesis that the shortest lunch lap possible is 400 metres. In this lesson they investigate how it might be possible to prove that a distance is the shortest possible, using a simplified problem.

### Lesson 3: Reflections

Students observe that the shortest lunch lap forms a parallelogram, justify this using symmetry, and show that for any rectangle of given length and width the shortest path is always twice the length of the diagonal.

Last updated September 14 2020.