Welcome 
Welcome to the eleventh 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

Many students find constructions a difficult topic to grasp. Perhaps it is the addition of extra equipment (compasses, rulers, angle measurers), or the unstructured nature of some of the questions. Alternatively it could be the case that, like me, some students just find such visual topics tricky to master. Dynamic geometry software should never replace paper and pencil. Until exams are set on computers, there will always be a need for students to be able to do things by hand. Perhaps more importantly, as this is how the great mathematical discoveries were made, students should be given the opportunity to see their beauty. But dynamic geometry certainly does have a role to play. It allows us to look deeper at each construction; to instantly change variables, look at why something works, and take one step closer along the road to generalisation. 



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 Bisector 
Have a look at this construction to create an angle bisector. Can you explain why it works? 

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 this topic require the full version of Autograph. 

Idea 1 – Perpendicular Bisector 
Download
1. Perpendicular.agg



The Autograph page shows a possible construction for the Perpendicular Bisector 

• 
Drag the nose of the vector to change the size of the circle arcs 
• 
Drag the circled points to change the angle of the line 
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Can you construct this on paper using compasses, ruler and pencil? 
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Can you make this construction from scratch on Autograph? 
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Can you explain why this construction works? 
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When does this construction not work? 
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Can you think of any other way to construct a perpendicular bisector? 

Idea 2 – Constructing Quadrilaterals 
Download
2. Quadrilateral Construct.agg



The Autograph page shows six quadrilaterals cunningly disguised to look like squares 

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Experiment by dragging the corners of the shapes around to see how they change 
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Can you identify which quadrilateral each shape actually is? 
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What are the properties of each quadrilateral? 
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Can you construct these quadrilaterals on paper using compasses, ruler and pencil? 
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Can you make these constructions from scratch on Autograph? 
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Can you explain why these construction work? 
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Which are the most difficult to construct? 
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Can you do the same with different types of triangles? 
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Note: If you want to see the secrets behind how each of these shapes were constructed, go to Object > Unhide All. 

Idea 3 – Alternative Angle Bisector 
Download
3. Alternative Angle Bisector.agg



The Autograph page shows a an alternative way of constructing the Angle Bisector 

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Drag the circled points to change the arcs of the circles 
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Can you construct this on paper using compasses, ruler and pencil? 
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Can you make this construction from scratch on Autograph? 
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Can you explain why this construction works? 
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When does this construction not work? 
• 
How does this relate to the more traditional way of constructing the angle bisector? 

Idea 4 – Triangle Construction Practise 
Download
4. Triangle Construction.agg



The following Autograph file is useful to use for checking students' attempts at constructing triangles 

• 
Set your students a challenge. For example:
– Construct a triangle with lengths 5cm, 7cm and 9cm
– Construct a triangle with lengths 8cm, 6cm and the angle in the middle of 50° 
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You can then build the triangle on Autograph and check the actual sizes of the other lengths and angles to measure your students' accuracy 
• 
To hide or reveal the lengths of the sides, drag the black boxes over the text 
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To hide or reveal the sizes of the angles, simply click on the angle arc and choose or remove Show Label 
• 
Challenge your students to come up with sets of lengths and angles that result in triangles that are impossible to construct 
• 
Which set of three criteria result in unique triangles and which allow you to create more than one triangle? 




Video Tutorials

The following video takes you through, stepbystep, how you can construct a circle through any three points. 




Handy Autograph Tip

The key to many geometrical constructions in Autograph, especially when creating the various quadrilaterals, is a Parallel line. Here is one way you could make one:



Open Autograph in Standard Mode 

Make sure you are in Whiteboard Mode 

Place two points anywhere on the page fairly close together 

Drag the cursor around both of these points so that they are both selected. Rightclick and choose Straight Line from the menu


Place another point somewhere on the page 

Make sure just the line is selected, and this new point are selected. Rightclick and choose Parallel Line from the menu.


Dragging the circles around the page allows you to change the gradient of both lines, and dragging the new point around allows you to change the position of the parallel line. Challenge: Can you use this as a starting point to construct a parallelogram? 




