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Autograph Newsletter 11 – Constructions

May 26, 2014 - Autograph, Autograph Newsletters
Welcome
Welcome to the eleventh 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.
Contents
Introduction
Diagnostic Question
Free Online Autograph Activity
Ideas for Extension
Video Tutorials
Handy Autograph Tip
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.
Reflection Diagnostic Question
Free Online Autograph Activity
Angle Bisector
Have a look at this construction to create an angle bisector. Can you explain why it works?
Angle Bisector
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
Can you construct this on paper using compasses, ruler and pencil?
Can you make this construction from scratch on Autograph?
Can you explain why this construction works?
When does this construction not work?
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
 Experiment by dragging the corners of the shapes around to see how they change
Can you identify which quadrilateral each shape actually is?
What are the properties of each quadrilateral?
Can you construct these quadrilaterals on paper using compasses, ruler and pencil?
Can you make these constructions from scratch on Autograph?
Can you explain why these construction work?
Which are the most difficult to construct?
Can you do the same with different types of triangles?
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
 Drag the circled points to change the arcs of the circles
Can you construct this on paper using compasses, ruler and pencil?
Can you make this construction from scratch on Autograph?
Can you explain why this construction works?
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°
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
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, step-by-step, 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. Right-click 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. Right-click 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?

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