1. GeoGebra Conjecture & Proof Mark Dawes
Comberton Village College University of Cambridge

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3.
Some set-up might be needed:

4. Question: “Make a conjecture about the triangle in a semi-circle
Question: “Make a conjecture about the triangle in a semi-circle. Prove it.”

5. John’s Conjecture

6. My expectation … y x y x This is what I expected: Label angle ACE = x
Triangle ACE is isosceles, so angle AEC = x
Label angle ADE = y
Triangle ADE is isosceles, so angle AED = y
In triangle DEC we have x+x+y+y = 180, so x+y = 90 and angle DEC is therefore a right angle.
This is the proof I learned at school, and while it works and certainly proves that the angle is 90 degrees, it is not very satisfying because it doesn’t really explain what is going on.

7. John’s Conjecture
The two smaller triangles have the same area

8. Maddie’s Diagram

9. Use materials at https://www. geogebra
Use materials at Find the ‘book’ named for this session.

10. Find the circle A circle passes through the x-axis at 2 and 8.
It is tangent to the y-axis.
Where is the centre of the circle?

11. Homework
“Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?”

12. Homework
“Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?”

13. Homework
“Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?”

15. Skipping 7,2 We need a regular polygon.
I choose to start with a regular heptagon.
I then chose to use a ‘skip’ of 2. I started at the top vertex, went clockwise round the heptagon two vertices.
If I continue to use a skip of 2 it eventually gets back to the starting point. I have created this diagram.
Explore what happens for different starting polygons and skip values.
7,2

16. Sliding Ladder
A ladder 10m long (!) rests on horizontal ground against a vertical wall. As the foot of the ladder slips away from the wall, what is the locus of the centre of the ladder?

17. Quadrilateral Tessellation
Claim: “Every quadrilateral tessellates.”
Is this true?
Can you explain why (not) ?

18. Which is bigger? Draw a point on a square.
You can make a new square by rotating this point about the centre of the square.
You can make a new rectangle by reflecting this point in the diagonals of the square.

19. Conjecture & Proof with GeoGebra
The Question
Instructions
Draw the diagram
Solve the problem
Record results/ideas/answers
Write a proof
Decide whether dynamic geometry is worth using
Are these steps in the right order?
Are there any steps missing?