Welcome 
Welcome to the sixteenth Autograph Newsletter! Each jampacked edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better. 






Introduction 
An often underused aspect of Autograph is its 3D capabilities. This is a crime against mathematics, as Autograph’s 3D engine is unique in the world of dynamic geometry in the way it allows students to visualise key concepts, manipulate objects, make predictions and test them out. This spans the word of graphing, geometry and calculus, meaning that benefits can be enjoyed by students of all ages and abilities. In this newsletter we will look at some of the many uses of Autograph in 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 
Discovering the Conic Sections 
Where do the conic graphs come from? Let’s find out! 

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 Development 
The following ideas for developing the topic of threedimensions require the full version of Autograph. 

Activity 1 – Transformations in 3D 
Download 1. Transformations in 3D.agg 


Did you know it’s possible to carry out rotations in 3D? 

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To rotate in 3D we need a line, not a point. Can you explain why? 
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The blue object has been rotated 90° clockwise about the line x=1 to give the green object. 
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If you wiggle one of the corners of the original object, can you predict which corner of the image will move? 
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Use the Animation Controller to see what happens as you change the angle of rotation. 
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To try out another rotation, add a point, rightclick and choose a line parallel to one of the axes. 
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Make sure the line and the object are selected, rightclick and choose rotation from the menu. 
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Experiment to your heart’s content with rotations in 3D! 
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Challenge: Can you figure out how to carry out reflections and enlargements in 3D? 


Activity 2 – Pythagoras and Trigonometry in 3D 
Download 2. 3D Trig.agg 


Can your students work out the length of the diagonal of this 2x2x2 cube? 

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Use Drag to take a look around the cube. 
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How can we work out the length of the diagonal line? 
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Are there any right angled triangles that can help us? 
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Create a line segment to help students see the appropriate rightangled triangle by selecting the points at two corners, rightclick and choose Line Segment from the menu. 
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To check students answers, click View > Status Bar and select the line you want to measure. 
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If you also want to challenge the students to work out the value of the angles, select two line segments, rightclick and choose Angle Between Lines. 

Activity 3 – Volume of Revolution 
Download 3. Volume of revolution.agg 


The way Autograph brings the concept of volume of revolution to life in 3D is quite something! 

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Challenge your students to picture the solid that will form if we rotate the shaded area 360° around the xaxis 
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When you are ready, select Slow Plot, leftclick on the shaded area, rightclick and choose Find Volume. 
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Hold down leftclick and drag the cursor around the screen to see the solid emerge 
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Now you can challenge your students to work out the volume. 
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A good way to start is to select the point at x = 3 and drag it closer to the other point (see Handy Autograph Tip below). 
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Students should hopefully see that the solid begins to resemble a cylinder the closer it gets, and following on from this the entire volume itself can be thought of as the sum of lots of these thin cylinders. 

Activity 4 – Vector Golf 3D 
Download 4. Vector golf 3d.agg 


What’s better than 2D Vector Golf? 3D Vector Golf, of course! 

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Your ball is the point at (4, 3, 2), and the hole is the centre of black cube at (4, 0, 3) 
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There are plenty of obstacles in your way, which you are not allowed to hit, and you have to try to get the ball into the hole using a few shots as possible 
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Here’s how you take a shot:
– Leftclick on an unoccupied part of the page to make sure no objects are selected
– Leftclick to select the current position of your point
– Leftclick on the vector (or vectors) of your choice – the one(s) you select should turn black
– Rightclick and choose either:
Copy Vector – if you have just selected one vector and you want to travel along it
Negative Vector – if you have selected just one vector and you want to travel along the negative of it
Multiply Vector – if you have selected just one vector and you want to travel along a multiple of it (either positive or negative)
Add Vectors – if you have selected more than one vector and you want to add them together
Subtract Vectors – if you have selected more than one vector and you want to subtract them. Note: When subtracting vectors, the order which you select the vectors is very important. The second vector you select will be subtracted from the first. 
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What is the fewest number of shots you need? 
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Challenge your students to design some nightmare 3D golf holes! 




Video Tutorials 
The following video takes you through, stepbystep, how to construct the 3D cube that has been used a few times throughout this newsletter. 




Handy Autograph Tip 
Seeing as this is a special 3D newsletter, I have two lovely little tips lined up for you. 

Tip 1
If you want to look at the properties of shapes in 3D, for example to look at their symmetry or number of edges, you can very easily access some premade ones. 


Open Autograph in Standard Mode and click on New 3D Graph Page. 

Click on Enter Shape. Click on Presets and choose a shape. 

Make sure you are in Select Mode. 

You can now drag the page around and have a good look at the properties of this shape. Clicking on the centre of the shape allows you to drag it to different locations. Clicking on one of the vertices allows you to drag that to a new position and hence alter the properties of the shape. 


Tip 2
The ability to zoom in and out on Autograph in 3D comes in incredibly handy, and it is very easy to do! 


Using the shape you created above. 

Make sure you are in Select Mode. To zoom in and zoom out, hold down Ctrl, hold down leftclick and drag the mouse forwards and backwards! 


