Free Dynamic Mathematics Resources
 5. Rotations
 Welcome Welcome to the fifth Autograph Newsletter! Each jam-packed edition will look at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better. Hope you enjoy it! Craig Barton Advanced Skills Teacher, creator of www.mrbartonmaths.com and TES Secondary Maths Adviser. Follow me on Twitter: @TESMaths
 Contents Introduction Diagnostic Question Free Online Autograph Activity Ideas for Extension Video Tutorials Handy Autograph Tip
 Introduction Following on from Newsletter 1 on reflection, here we turn our attention to another of the transformations - rotation. For many students, this is the most difficult of the transformations, and I can certainly sympathise. I find rotations difficult to picture in my head and interpret on paper. Fortunately, there is dynamic geometry software such as Autograph to help students (and me!) out. The ability to manipulate objects and points, immediately change the angle of rotation, and hide crucial information, allows us to put together a package of resources and activities to really bring the topic of rotation alive. We can even venture into the world of 3D!
 Diagnostic Question Diagnostic questions are ideal to use at the start of the lesson to enable you to get a quick and accurate picture of your students' levels of understanding. They are designed in such a way that common misconceptions that your students may hold should steer them to one of the incorrect answers, thus allowing you to learn where the problems lie from their responses. Typically I give my class 30 seconds thinking time and then ask them to hold up their fingers: 1 for A, 2 for B, etc.
 Free Online Autograph Activity Rotation Golf! Can you change the centre of rotation and the angle in order to fit the red object into each of the black holes? Describe the rotation in each case. These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
Ideas for Extension
The following ideas for extending the topic of reflection require the full version of Autograph. Click on the image to download the individual Autograph files.
Idea 1 – Find the Centre
Turn the topic of rotation on its head by challenging students to find the three centres of rotation
 • The blue object has been rotated about three different centres of rotation by 90, 180 and 270 degrees clockwise to produce the three images. • Can you work out the three centres of rotation? • Use the Scribble Tool to mark the students' predictions. • If students are struggling, move the blue object to the right one square and look at the new positions of the images. • To reveal the centres of rotation, click the Object menu and choose Unhide all. • If I now move the blue object up one square, how will the rotated images move? • You can set up more examples like this using the techniques in the video below.
Idea 2 – Rotation Patterns
Can your students make nice patterns using rotations?
 • Can your students see how my pattern has been put together? • Try moving one of the shapes on the right and see what happens. • Can your students make a pattern using just one shape? • How about using two or three? • Why do some patterns have gaps and others have no gaps? • If your students want to share their patterns, email support@autograph-maths.com. • The techniques to carry out the rotations are explained in the video below.
Idea 3 – Combining Rotations
What happens when we combine two rotations?
 • The blue object has been rotated 90° anti-clockwise about point (1, 3) to give the purple object. • To purple object has then been rotated 90° anti-clockwise about point (5, 2) to give the green object • Can you think of a single rotation that would go directly from the blue object to the green object? • Carry out the rotation (using the techniques described in the video) to check your answer. • Investigate if it is always the case that any two rotations can be represented by a single rotation. • Can you find a way of doing this without using the graph?
Idea 4 – Rotations in 3D
Did you know it's possible to carry out rotations in 3D?
 • To rotate in 3D we need a line, not a point. Can you explain why? • The blue object has been rotated 90° clockwise about the line x = 1 to give the green object. • If you wiggle one of the corners of the original object, can you predict which corner of the image will move? • Use the Animation Controller to see what happens as you change the angle of rotation. • To try out another rotation, add a point, right-click and choose a line parallel to one of the axes. • Make sure the line and the object are selected, right-click and choose rotation from the menu. • Experiment to your heart's content with rotations in 3D!
 Video Tutorials The following video takes you through, step-by-step, some of the ways you can use Autograph to look at the topic of rotations.
Handy Autograph Tip
You can save yourself a lot of time creating a shape by using marquee select, but only if you are careful!
 Open Autograph in Standard Level Make sure you are in Whiteboard Mode Place four points on the page in a clockwise direction to create a quadrilateral In Select Mode drag around all four points, right-click and select Group to Shape from the menu You should have a lovely shape Now try the same thing again, but this time when you create the quadrilateral, place the points at opposite corners first Select these new points in the same way and group the shape, and you should end up with something that resembles a bow-tie! The lesson to be learnt from this, is be careful the order you place your points!
 Curriculum Coverage Not teaching rotations today? No problem, Autograph covers much more including statistics, probability, graphing, vectors, coordinate geometry, shapes and transformations, numerical methods, cartesian equations, parametric equations, polar equations, differential equations, calculus, trigonometry and all in two- and three-dimensions. Douglas Butler's series of 7 Autograph Tutorial Videos provide a perfect overview of curriculum coverage.
 Free Autograph Activities If you need free Autograph Activities that can be used without Autograph installed try Mr Barton's Autograph Activities.
 Getting Started If you are just getting started with Autograph try the Video Training Series and the Training Manual with step-by-step instructions.
 Worksheets Autograph trainer Alan Catley has created an impressive collection of student worksheets which can all be downloaded and used for no charge.
 Video Tutorials Watch videos by Craig Barton, Douglas Butler, Hassan Lakiss and Janet Smith.