Using Autograph to Teach Calculus topics from AS and A2 Level Mathematics and Further Mathematics

1. Using ‘Autograph’ to teach
Calculus topics from AS and A2
Level Maths and Further Maths.
Darren Barton – Southampton University
16th January 2012
d.barton@poolehigh.poole.sch.uk
tinyurl.com/amaths
@mrdebarton
16/01/2012 Darren Barton – Southampton Uni – 16/01/12 1

2. Introduction to Workshop
• Five mini-sessions of 15 minutes each.
1.) Sketching and Transforming Graphs.
2.) Numerical Integration.
3.) Volumes of Revolution.
4.) Differentiating Trig Functions.
5.) Sketching and Differentiating y = ax.
• Short presentation (5 mins).
• Hands on experimentation (10 mins).
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3. 1.) Sketching and Transforming Graphs
- Cartesian.
- Polar.
- Dynamic constant editor.
- Slow plotting.
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4. Can you draw the ‘Maths Man’?
What three plenary questions could you
ask students to summarise their learning?
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5. Extension!
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6. Polar Graphs
Use Autograph to experiment with
drawing graphs of the form
r = a + b sin (cθ) where a, b, c ϵ Z
Can you sketch these graphs?
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7. 2.) Numerical Integration
- Use Autograph to establish principle of
‘Trapezium Rule’ etc.
- y = x2 + 1 with domain 0 ≤ x ≤ 1 and range
0 ≤ y ≤ 2.
- Two intervals?
- Over or under estimate?
- How do you improve the accuracy of the
approximation?
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9. 3.) Volumes of Revolution
- Visualising 3D shapes.
- Using ‘Jing’ as a teaching tool.
- Watch the hyperlinked clip.
- http://screencast.com/t/0lkBztEAxvp
- Now try yourself!
- Can you drawn a ‘rugby ball’?
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10. Visualising an approximation of the volume of revolution generated
when y = x2 is rotated 2π about the x-axis using rectangular strips
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11. Can you use Autograph to drawn the
‘rugby’ ball?
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12. 4.) Differentiating Trigonometric Functions
- Plot y = sin x.
- Plot dy/dx.
- Describe the relationship between the curves.
- Can you explain why?
- Can you hypothesise the relationship between
y = cos x and the associated dy/dx curve?
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15. Investigate
Sketch the graphs of y = sinh(x), y = cosh(x)
and their associated derivatives.
Can you derive graphically the rules for the
derivatives of y = sinh(x), y =
cosh(x), y = sinh(ax) etc.
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16. 5.) Sketching and Differentiating y = ax
- Sketch y = ax.
- Sketch dy/dx.
- What happens as a → e?
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