Classroom intervention Grade X: Geometry

The problems:
1) Where do I start?
As I went through what they had for Geometry I realized that although they had done coordinate geometry for a couple of yrs, trigonometry for a couple of yrs, they were now going to encounter the idea that triangles can be similar and their sides would hold ratio between them! I also noted that there was no proof given in the text for the same :). Great, I can start with one of the most powerful concepts in geometry that allows me to link with everything they kida, sorta know of and I don't even need to prove it.

2) The kids may remember the concepts but not the corresponding names e.g. adjacent angles, corresponding angles. They would, of course, remember vertically opposite angles, but who doesn't. This gives rise to the issue that we can't talk the same language.

3) The kids also know me to generally have fun with puzzles and games and I need to have some connection with writing examinations.

Class 1
I started with asking kids to make a cheat sheet  (bit paper as the kids call it) for Geometry by putting in all that they know about it. The kids were excited by the prospect though I made it clear I was not encouraging them to take it to examination.

We started with a small list which was enough for our purposes and decided to add more as we went along:

I added one new one of similar triangles, with just one rule (AA) if two angles of two triangles are equal the triangles are similar. In similar triangles there is a constant ratio between any two corresponding sides.

I mentioned to them that the similarity of triangles is the most powerful of the geometrical theorems and now they can rule the world. We started with trying to prove the 'mid-point formula'.

Actually, if you can understand the picture the steps are simple, but I went through many iterations of building up this picture one piece at a time. The biggest bottleneck was that they didn't associate the coordinate in coordinate geometry with a real distance. In a point (x1,y1) they were unclear as to what x1 really was. It had just been a number they needed to substitute in an equation. Well, this was precisely what I wanted to address and I didn't mind going over it again and again. Once that is understood it was actually quite easy for them to take up the new concept of ∆CBD ~ ∆CAE and were even able to supply be the reasons (adjacent angle, right angle). As 

CA/CB=2 (mid point) 

 (x2-x1)/(x2-x)=2

=> x2-x1 = 2*x2 - 2*x 

=> x2-x1-2*x2 = -2*x

=> x2+x1 = 2*x or x = (x1+x2)/2

I proved it for x and I asked them to prove it for y at home. I completed the picture as below.

I asked them to complete a couple of problem sets in the book and using similar triangles prove Thales Theorem as well.

I worked out a couple of questions from two exercises and asked them to do the rest by themselves.

Class 2

I didn't expect everyone to do everything, but I expected someone to do something, and they did.

Four kids (in a class of 16) tried and succeeded in proving the mid point theorem, few others attempted the exercises in the book, as they now looked trivial. Few read the book by themselves for the first time to see what's really there in it :). However, there was still half the class that was waiting for the dust to settle and for me to solve the questions on the board.(ha)

I proved Thales theorem using similar triangles and asked them to add this to their cheat sheet as well.

I asked them to continue to solve the problems in their book. It didn't matter that they had done half the exercise or looked sheepish that they hadn't started. They all had to work and work independently.

Its surprising how noisy the classroom usually is, its not the kind of organized chaos the teachers permit its the kind that comes from boredom of kids who are able to solve faster and waiting time of those who don't follow for the answer to be written on the board.

They could not talk or borrow pencils/pens/erasers/scales or any of the other million things they seem to always need in the classroom. The instruction was, use what you have, do what you can and find out for yourself what you are capable of. It was not easy, they had not done this except in examinations. I needed a more supportive setting than an exam and I gave occasional individual attention, at times needed to remind a couple of kids about what we are doing, but we were able to get some peace and quiet in the class (another first). I went around to see how kids were doing and for once compromised and let kids know when they had got the answers. Almost 4-5 kids other than the ones who had already done some work were very kicked and would say, is that all there is to it.

This class was a very big achievement, kids worked and to their ability succeeded in solving some problems and started believing in their ability to do so. The most questions solved were close to 10, but even the least were 3. Usually 4-5 questions are covered in class when the teacher solves it on the board, not bad.

Class 3

It was much easier this time for the kids to settle and work this time. I addressed a couple of common issues encountered in the previous class. We talked about the common strategy in the questions to have multiple triangles and a common side that was divide in the same ratio as the others. We addressed ideas for cute questions like ratio of sides of a trapezium that needed an additional line to be drawn to see the triangles.

We also compared altitudes of similar triangles and proved that they are the same ratio (of course, using the idea of similar triangles). This brought us to right angled triangles and all the physics they had done of comparison of shadows of objects and their heights. We talked about how these triangles are the same and what would happen if the time or place of one measurement is different from another and what the assumption is to say that the angle of incidence is the same. We concluded that the rays of the sun need to be parallel between the two objects, times and places of the measurement. Our misconceptions are buried deep, so for yucks I drew the sun, as we drew as kids with rays coming out in all directions and pointed out that this in such a picture the rays don't seem parallel and any bright objects gives light in all directions :). It was a fun, after a couple of intense classes this whole exchange seemed just what we needed.

We then cover altitudes of right angled triangles that create smaller right angled triangles that (needs rotation carefully) are similar to the original triangle, which further can have altitudes giving similar and smaller right angled triangles. 

This was a fun exercise and we really needed to be sharp about our rotations. We finally figured out the algorithm to get things right. Start with the corner where the angle is unchanged, mention the corner with the right angle next and finally the corner left out.

I found a really cute problem in the book. Try it out:

Find the other diagonal AC given AB=BC, AD=CD and the right angled triangles marked as above.