Rotation Reflections & Proof
'Can you produce two reflections in a single rotation? Prove it!'
Many of us did paint reflection art to create a pretty butterfly when we were in elementary/primary school. Since then, equations of lines have replaced the folds in a piece of paper (or a mirror) and centres of rotation, degrees and direction have replaced "turning it on its side"! This is a great investigation that explores all these new complexities in the world of reflections and rotations, and furthermore offers a really satisfying opportunity/introduction to the holy grail of mathematics - proof.
The video below explains the task, followed by the applets/simulations you're going to be using. There is also a paper&pencil print-out if you would rather use tracing paper to investigate the transformations (see the "Help" section below).
Students need to be familiar with equations of lines and defining rotations and reflections. If students need a review of these skills they can try these activities: equations of lines and reflections Equation Reflections, defining a rotation Rotation Navigation
Presentation / Note Taking software: Open-Sankoré (FREE, open source, presentation software for computers and whiteboards), Notability (for iPad: $3.99) and Protractor (many other protractor apps available), OneNote (included with Microsoft Office or FREE download)
Firefox and Internet Explorer web browsers should automatically install the necessary player so that you can view the applets when you click on the text or images below. If you are using google chrome, or for some reason the player is not working in your browser, you can install it very quickly from here.
1. Horizontal and Vertical Line Reflection Rotation Investigation
2. Diagonal Line Diagonal Line Reflection Rotation Investigation
The teacher, or some students, may prefer to do this activity using pencil, paper and tracing paper. The "Help" section below provides a video for students to watch with a few hints & tips for reflections and rotations using paper & tracing paper (oven/grease-proof paper is a good, cheap alternative for tracing paper)
- Printable version of the same.
Pencil and Paper rotations and reflections: hints and tips
Advantages of the activity
- This activity effectively brings together knowledge from a number of topics: equations of lines, rotations and reflections, angle theorems and congruent triangles.
- The task can be very simply stated: "Can you find a single rotation that has the same effect as two reflections?".
- The focus of the activity is on students' investigation skills: experiment (using the applets/simulations above), record their results (on paper or using annotation software), organise & analyse their results in the search for any patterns. If they find any patterns they can then form a hypothesis/conjecture, test this conjecture against further examples (that haven't yet been tried) and finally . . prove it?