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Autograph Activities Free Dynamic Mathematics Resources
16. 3D Special
Welcome to the sixteenth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
Diagnostic Question
Free Online Autograph Activity
Ideas for Development
Video Tutorials
Handy Autograph Tip
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.
Diagnostic Question
Free Online Autograph Activity
Discovering the Conic Sections
Where do the conic graphs come from? Let’s find out!
Discovering the Conic Sections
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 three-dimensions 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?
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!
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?
Use Drag to take a look around the cube.
How can we work out the length of the diagonal line?
Are there any right angled triangles that can help us?
Create a line segment to help students see the appropriate right-angled triangle by selecting the points at two corners, right-click and choose Line Segment from the menu.
To check students answers, click View > Status Bar and select the line you want to measure.
If you also want to challenge the students to work out the value of the angles, select two line segments, right-click 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!
Challenge your students to picture the solid that will form if we rotate the shaded area 360° around the x-axis
When you are ready, select Slow Plot, left-click on the shaded area, right-click and choose Find Volume.
Hold down left-click and drag the cursor around the screen to see the solid emerge
Now you can challenge your students to work out the volume.
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).
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!
Your ball is the point at (-4, -3, -2), and the hole is the centre of black cube at (4, 0, 3)
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
Here’s how you take a shot:
– Left-click on an unoccupied part of the page to make sure no objects are selected
– Left-click to select the current position of your point
– Left-click on the vector (or vectors) of your choice – the one(s) you select should turn black
– Right-click 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.
What is the fewest number of shots you need?
Challenge your students to design some nightmare 3D golf holes!
Video Tutorials
The following video takes you through, step-by-step, 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 left-click and drag the mouse forwards and backwards!
Craig Barton
Craig Barton
Advanced Skills Teacher, creator of and TES Secondary Maths Adviser. Follow me on Twitter: @TESMaths
Curriculum Coverage
Autograph covers 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.
List of Autograph Activities
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.
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.
Past Newsletters
Read other newsletters in this series by visiting the newsletters page.
Prefer printed material? Download sample chapters from the Autograph Activities books.
Autograph Activities Books
New this month: Dynamic Statistics with Autograph and Dynamic Algebra with Autograph by Dr James Kett.
Courses and Trainers
If you prefer face-to-face training attend a training course or book a trainer to come to your school.
Be Inspired
Been using Autograph for a while and looking for new ideas? For inspiration try Owen Elton's Matheminutes blog, and the blog from Autograph's creator Douglas Butler.
Jonny Griffiths has three resources sites which make use of Autograph: Rich Starting Points for A Level Core Mathematics, Making Statistics Vital and Carom Maths.
More Resources
A complete list of useful websites can be found on Douglas' TSM Resources website.
You can find many of these resources on the Autograph TES page.
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