Autograph Newsletter 14 – Geometric Proof

Welcome to the fourteenth 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
Further Activities
Video Tutorials
Handy Autograph Tip
Just like in Newsletter 12, in this edition we draw our inspiration from the wonderful Median maths blog by Don Steward. Don’s activities challenge students to investigate, make conjectures and solve problems, and they have formed the basis of some of my most enjoyable lessons. Several of the activities on Median revolve around geometric proof. This is a topic that many students can have difficulty with, possibly due to its abstract nature, and being able to see the whole picture in terms of the steps that are required to solve the problem. Here is where I feel Autograph can help. By setting up the problems and then playing around with them dynamically, students may be able to see links and relationships that were previously hidden, make further conjectures, and test out their proofs.
Diagnostic Question
Usually we have a Diagnostic Question in this spot, but for this special edition of the newsletter we are instead I wanted to show you the kind of activity Don has on his website. Often these are images which can be copied and used directly, or pasted onto a PowerPoint slide or Word file. Here is a nice way to introduce students to the word of geometric proof using angles in triangles.
Box Plot
Free Online Autograph Activity
Can you Prove it 3?
A tricky old construction here that leads to the surprising result that these two angles appear to be always equal. The question is… can you prove it?
Can you Prove it 3?
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.
Further Activities
The following ideas for activities are also taken from Don Steward’s website. Try them on paper first and then turn to Autograph to look at them in more depth. Click on the image to download the individual Autograph files.
Activity 1 – The Equilateral House
Download 1. Equilateral House.agg
Challenge: A square has an equilateral triangle constructed on each of two of its adjacent sides. The top corner of each triangle is connected to each other and to the far corner of the square. Can you prove that the resulting shape is an equilateral triangle?
The situation has been modelled on Autograph
 Experiment by moving the positions of the three corners of the triangle around the page
Convince yourself that the triangle is equilateral by measuring the sides or the angles:
– Measure sides: click on a side and the distance will be displayed in the status bar at the bottom of the screen
– Measure angles: double click on the angle arc and place a tick by Show Label
Does this help you prove that the triangle is equilateral?
Activity 2 – The Mysterious Yellow Triangle
Download  2. Mysterious Yellow Triangle.agg
Challenge: Draw any right angled triangle, bisect an angle and construct the perpendicular to the hypotenuse (an altitude). Can you prove that the yellow triangle is isosceles.
The situation has be modelled using Autograph
 Experiment by dragging the three corners of the right angled triangle around the page
Does this help you prove that the yellow triangle must be isosceles?
Activity 3 – Isosceles Split
Download  3. Isosceles Split.agg
Challenge: What kinds of triangles will split into two (non-congruent) isosceles triangles? The split must be a straight line from one corner of the triangle to an edge. Don Steward claims there are three different types of triangle for which this can be done, each with a set of angles that have a special property. Can you find them?
Once again the situation can be modelled using Autograph
To change the shape of the triangles, simply drag their corners around
Test out your students predictions and variations for types of triangles
Investigate why some work and others do not
Activity 4 – The Special Parallelogram
Download  4. The Special Parallelogram.agg
Challenge: Normally when the two base angle of a parallelogram are bisected, the resulting lines do not meet on an edge of the parallelogram. When does this happen? Can you prove it?
This problem in particular lends itself well to being dynamically modelled and investigated using Autograph
 To change the shape of the parallelogram, simply drag the corners. You will notice that the sizes of the angles change as well
As a hint towards (one possible) solution, the intersection of the two bisectors has been marked with a point. You can create and measure the angle between these two lines as follows:
– Click on the three points that define the angle in order
– Right-click and select Angle from the drop-down menu
Video Tutorials
The following video takes you on a short trip around Don Steward’s wonderful blog
Handy Autograph Tip
I really like the way Autograph does Constructions as the information you need to give it makes logical sense. If you select the correct objects, the construction you are looking for will be waiting patiently for you in the right-click menu. A useful activity might be to challenge the students to see if they can figure out how to create these constructions on Autograph
Angle Bisector: select the three points, in order, that make up the angle you want to bisect
Perpendicular Bisector: choose the two points you want to bisect
Parallel Line through a Point: select the line in question and the point you wish it to pass through
Perpendicular Line through a Point: exactly the same method as above
Can your students discover any others?…

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