Free Dynamic Mathematics Resources
 3. Angles
 Welcome Welcome to the third 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 Students are first introduced to angles at an early age, and they keep cropping up throughout all of their maths education. Starting with the introduction of degrees and how to measure angles, the topic progresses through angle rules, bearings, right up to the concept of radians and trigonometric identities. Dynamic software, like Autograph, can help with all aspects of the topic of angles. By making the most of the ability to instantly manipulate things, as well as hiding and revealing information, we can hunt for relationships, test them out and work towards proofs. The days of drawing hundreds of examples on the classroom board have long gone! Hopefully you will find lots of ideas and resources to help with the teaching and learning of angles in this newsletter.
 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 Angle Facts Can you spot each of the key angle facts using this interactive activity? Change the slope and position of the lines and see if they are still true. 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 straight line graphs require the full version of Autograph. Click on the image to download the individual Autograph files.
Idea 1 – Make the Angle
Students are challenged to create three angles of given sizes, and then use Autograph to check the accuracy of their attempts.
 • What strategies did you use? • Which angle was the most difficult to make? • How do you know when an angle is greater than 90°? • How about greater than 180°? • How far off where your guesses? • Use the Scribble Tool to mark the students' predictions and any workings out • Note: You can change the target angles to whatever you like, and to reveal the size of the angles, simply double-click on the angle and place a tick by Show Label
Idea 2 – How big is the Angle?
Extending the previous idea by this time asking students to estimate how big a set of angles are.
 • What strategies do you use when estimating the sizes of angles? • What size do you know your angle is definitely greater than? • How about definitely smaller than? • Can you think of a clever strategy for estimating angles greater than 180°? • Again, make use of the Scribble Tool to mark the students' predictions and any workings out • Note: To adjust the size of the angles, simply drag the ends of the arms of the angles into new positions
Idea 3 – Bearings
Use this Autograph file to introduce students to the concept of bearings, and to help them spot relationships between bearings.
 • What do you notice about the direction bearings are always measured from? • If you know the size of one of the angles, how can you work out the size of the other? • Do angle facts help at all? • Note: Autograph does not give single and double-digit angles three figures
Idea 4 – Angles between lines in 3D
We can even enter the world of 3D to calculate the angle between two vector equations of straight lines
 • To change or view the vector equation of the lines, simply double-click on either line • To work out the angle between the two lines, make sure both lines are selected, right-click and select "Angle between lines" from the menu • The angle then appears in the Results Box • Can you give the vector equations of two lines with a 90° angle between them? • How about a 45° angle?
 Video Tutorials The following video takes you through, step-by-step, some of the ways you can use Autograph to measure angles
Handy Autograph Tip
Autograph has a particular way of measuring angles, and unless you know it you might come un-stuck, especially when trying to measure angles greater than 180°!
 Open Autograph in Standard Level Make sure you are in Whiteboard Mode Click on No Axes. Place 3 points anywhere on the page. Note: if you know how you can join the points up with two line segments to form an angle as shown in the video above. Starting with the point on the far left of the page, and with nothing else selected, click on each point in turn in an anti-clockwise direction. Right-click and select Angle from the menu. Before clicking OK, make sure you have ticked "allow reflex angle" Now repeat the process, but this time click on your three points in the opposite order. You should find that one way measures a reflex angle and the other one does not. This is because Autograph measures angles in an anti-clockwise direction! You could of course use this demonstration for investigating the sum of angles around a point!
 Curriculum Coverage Not teaching angles 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.