Autograph Newsletter 9 – Pythagoras and Trigonometry

Welcome to the ninth 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 Extension
Video Tutorials
Handy Autograph Tip
A common perception is that you cannot use Autograph to aid the teaching and learning of Pythagoras and Trigonometry. The good news is… you can! In terms of work with 2D triangles, we can set up Autograph files that allow you to manipulate sides and angles and challenge students to work out the missing information. We can then enter the world of 3D to not only work out the length of diagonals, but crucially to visualise exactly what is going on. Then we have the graphs of the trigonometric functions, which allow us to solve equations and practise graphical transformations. All in all, there is plenty Autograph can help us with in terms of Trigonometry and Pythagoras!
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
Trigonometry – Practise Finding Sides
Use this activity to practise finding the length of missing sides using Sin, Cos and Tan
Trigonometry - Finding Sides
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 – Pythagoras in 2D
Download  1. Pythagoras.agg
This Autograph file allows you to generate as many examples as you like to challenge students to calculate the length of missing sides using Pythagoras’ Theorem
To reveal the length of the missing sides, simply drag the black rectangle over the blue box
To change the size of the triangle, just drag the corners to a new position
To change the accuracy, go to View > Preferences > General and then change the accuracy
Should you wish to extend this to also measure angles, just click on the three corners that define the angle in order, right-click and choose Angle from the menu
Idea 2 – Autograph Extra: Trigonometry
Autograph has an excellent Extras page that can help students see where the graphs of the trigonometric functions come from
To open the page, Click on File > New Extra Page > Trigonometry
Choose Degrees or Radians
Then adjust the size of angle θ using the right-left arrow keys and watch as the graphs appear
You can also look at transformations by adjusting the size of a and b
Examples of questions to ask:
– Can you explain why Sin(θ) is given by the height of the right angled triangle?
– Why does the graph of Cos(θ) start at 1?
– How does the graph of Tan(θ) relate to the circle?
– What happens to the graph of Tan(θ) at 90°? How does the circle help explain this?
– What is the relationship between sin, cos and tan?
– What is the relationship between degrees and radians?
– What happens when a is 2? Why?
– How do the graphs of Cosec, Sec and Cot relate to the graphs of Sin, Cos and Tan?
Idea 3 – Pythagoras and Trigonometry in 3D
Download  3. 3D.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
Idea 4 – Trigonometric Graphs
Download  4. Trig Graphs.agg
You can use Autograph to investigate the graphs of the trigonometric functions
The page shows the graph of y = sin(x) and
y = a sin(b(x + c)) + d
Challenge your students to predict what impact the values of a, b, c and d have on the shape and position of the graph
Once the students have shared their thoughts, use the  Constant Controller to adjust the values of the constants
To investigate the other two graphs, simply double click on the graphs themselves and change the “sin” to cos or tan
You can also use these graphs to illustrate the solution to trigonometric equations. For example, to show the solution to sin (2x) = 0.5:
– Click on  Enter Equation and type in y = 0.5
– Use the  Constant Controller to adjust the values of a, b, c and d
This can help students see how the type of equation affects the number of solutions for a given domain.
Video Tutorials
The following video takes you through, step-by-step, how to construct as right-angled triangle to be used in the study of Pythagoras and Trigonometry.
Handy Autograph Tip
In the first Extension Activity I used a useful technique to hide and reveal key information. This can be applied to any topic, for example working out lengths, angles, equations of lines, etc.
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 Line Segment from the menu
Select this line segment, and click on Textbox. Click on Edit Font and change the text colour to white and click OK
Add four points to the page in the shape of a rectangle
In Select Mode, drag round these points, right-click and choose Group to Shape. If you now drag this rectangle over the place where the Textbox is, the text inside showing the length of the line segment should be revealed! What other applications of this technique can you think of?

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