# Teacher Notes

This resource asks students to find the locus of the focus of a parabola if the parabola is rolled along the $x$-axis. Students will find it helpful to know the standard parametric form of a parabola. The task is structured via a series of mini-questions to help students reach the interesting conclusion that the focus follows the path of a catenary curve ($y=a\cosh \frac{x}{a}$). As the problem involves arc length, this resource could be used after How long is a piece of string?

### Preparation

To encourage students to approach each mini-question in turn, a set of question cards for each group could be useful, or a printout of the sheet containing all 6 questions.

If students have not studied conics then some prior discussion of what the focus is would be helpful.

### Possible approaches

Students could be shown the applet as a whole class before working in small groups on the problem, either with a sheet of mini-questions, or having the questions issued as each one is answered.

Some think-pair-share time could be used, especially if students have devices on which they can explore the applet, before a whole class plenary which could include some revision of work on parabolas.

### Key questions

• Can you think of any alternative ways to express a general point on a parabola?
• What do you already know?
• Can you find an expression for the $x$-co-ordinate?

### Possible extension

Creating the applet as suggested!