This activity brings together a range of ideas from Activity 1 and 2 using a combination of rotation and dilation matrices and a very powerful visual and algebraic approach. The activity gives function to the use of matrices and highlights how relatively complicated expressions can be determined very easily. Students connect many aspects of the Specialist Mathematics course in a single activity that is sure to engage students.
- Locus definition and construction in the plane of lines, parabolas
- Distance formula and locus definitions of curves in the plane
- Invariance of properties under transformation
- Coordinate and matrix representation of points and transformations
- Dilation, rotations and reflections and invariance properties
About the Lesson
In activity 1 and 2 students determined equations for a family of straight lines, used simultaneous equations to calculate successive points of intersection and an equation for the locus formed by these points. A strong relationship exists between these lines, points and locus consisting of a simple rotation and dilation about the origin. Students use matrices to explore the rotation and dilation of these points, lines and locus. In the extension section of the activity students explore the invariant properties of the conic section following the rotation, that is the integrity of the parabolic curve and its relationship with the focus and directrix is maintained. No prior understanding of a locus/conic is required.