I have used visualisation with many classes down the years, although I have got out of practice! The task below was suggested to me by some work I did at the recent ATM conference and concerns midpoints and triangles.
Spoiler images at the bottom of the post!
Visualising Midpoints
We are going to do a simple visualisation task. For this
task you may not do any physical drawings, at first. You may have your eyes
open or closed as you see fit.
- Imagine an infinite flat plane.
- A fixed point, A, appears.
- A second point, B, comes into the plane and is also fixed.
- Join the line segment AB.
- A third point C enters the scene. It is free to move.
- Watch as the point moves.
- Pause and describe what you see to me.
- Right, now join AC and BC to make a triangle. Remember that A and B are fixed, but C is free to move.
- Find the midpoint of AC.
- Find the midpoint of BC.
- Join them.
- Watch what happens to this line as C moves.
- Tell me the story of what you see happening.
- What can you say about the line?
- Tell me any conjectures you have.
- Can we prove them?
Try the task now, you may come up with questions I have not!
At this point I would allow the students to draw diagrams and then in pairs choose a conjecture to prove.
It is important that the questions are the students, but helpful hints toward parallelism, area or perimeter, may be needed.
My questions;
- Are the base and the mid-line parallel? Why?
- What is the area of the smaller triangle in terms of the first? Why?
- What is the perimeter of the smaller triangle in terms of the first? Why?
Extensions;
- Instead of midpoints split the lines AC and BC into the ratio 1:2 and ask all the same questions (Area is particularly fruitful).
- What about the ratio 2:3?
- What about the ratio x:y?
- What if we split one line in the ratio 2:3 and the other in the ratio 1:2?
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