To make the session more interesting, I divided the participants into 2 groups, and created a mini competition between the groups. Groups earn points through solving problems, showing effort in solving problems, or actively contributing to the discussion.

The session started with an introduction to shortest paths: why they are necessary or useful in the real life. In delivery, manufacturing, or other industries, sometimes shortening the distance can lead to huge amounts of money saved on transportation. Moreover, in atheletics and other activities, traveling through the shortest distance is optimal.

First, we discussed what the shortest path connecting 2 points is. After proving the triangle inequality, we proved that the shortest path is the straight line connecting the 2 points. However, we might be interested in quantifying the exact distance. To find this, we use the pythagorean theorem.

The proof of the pythagorean theorem involves calculating the area of a square with side length a+b in 2 ways: first, using s^2, or (a+b)(a+b)=a^2+2ab+b^2. Second, viewing it as the sum of the areas of 4 triangles and a square. Each of the triangles has area ab/2, while the square has area c^2. Equating these and canceling gives the desired result.

Then, the distance of the shortest path from (-2,2) to (2,1) is sqrt(17).

Now, we add in another condition: the path must touch the x-axis.

This might not seem easy to do, but after reflecting (2,1) over the x-axis to (2,-1), we see that the straight line connecting this to (-2,2) has the same length as a path from (-2,2) to the x-axis and then to (2,1). Pythagorean theorem gives us that this length is 5.

Then, the distance of the shortest path from (2,1) to (1,3) such that it touches both the x and the y axis.

This is essentially the same problem, except 2 reflections need to be applied. The ponit (2,1) can be reflected over the x-axis, and the point (1,3) can be reflected over the y-axis. The straight line connecting these again correpsonds to the shortest path.

2 more problems were discussed, each of them involving more creative ways ot reflect points and lines.

Overall, the participants really enjoyed the session, especially the group competition and points system, along with the rewards. It also provided an incentive to focus on something they might not totally enjoy, and in the process discovering the beauty of the underlying logic and patterns.

In the next session, I will try different styles of discussion by experimenting with different word choices, team sizes, discussion methods, and perhaps rewards.

For those who missed the session: https://youtu.be/dDw3Tag86Dk

These are the recordings to the session

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