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August 26, 2013

Class X intervention: Prove it!

Please read the other two posts
1) Why intervention?
2) First few classes
if you care for continuity.

I'm consolidating the key things that happened over the next two classes.

The next topic within geometry was on properties of chords, tangents with circles. For a transition from triangles to circles and I tried to connect up the two by showing them that the x coordinate of a cow going in a circle at constant speed is a sine wave :). This was way beyond the syllabus and had the teacher browse through the textbook to see where this 'stuff' was. It was a quick transition that touched on trigonometry and gave me a chance to plant a seed that I would water in my electronics class with them as I introduce AC signals...

In my preparation I realized that they were supposed to know some really neat stuff with chords and circles in 9th grade. An important one being given a chord any triangle it makes with any point on the circle (on one side of the chord) has the same angle. I tried proving it myself, I couldn't. I didn't find the 6th grade textbook either and finally went though the proof on khanacademy. It had a proof that the angle (subtended by the arc) with the center is double of the angle at any point. Nice, I didn't remember this.

In class I mentioned the theorem and the kids and went a step further and said that the angle with the center is double of the angle it makes, they said that they didn't know it.  It wasn't enough that they knew now they wanted it proved :). I could not believe these are the same kids who would take a formula without question and were afraid of proofs 5 classes back.

Its a cute proof that involves a specific case of a triangle made with the diameter to make an isosceles triangle and then generalizes it by splitting any angle as a sum (or difference) of two angles involving the diameter.

I proved the diameter and then gave an example of splitting a random angle as a sum of two angles. They were not convinced that this was general enough and came up with a point where the angles could not be added (point A - a limiting case of summation). I gave them the proof by subtracting the angles. The wheels in their heads were really whirling now. Then found a point on the other side of the chord (point B) and I told them that it was indeed not valid on this side o the chord and we would talk about it later.

There are only a few proofs that are going to come in the X class exam. The advantage was that I could still prove everything but didn't have to write it out in full on the board. We looked at the figure and I made my case by walking them through the thought process while marking on the figure. Its a real class and there are kids that zone out.

 The good part is that since I didn't need to write things out it doesn't take time to repeat the proof from where they didn't follow it. There may still be one child with a glazed look and when you ask him/her what he/she missed would come back with 'didn't understand anything'. In such cases you just start at the beginning, there is a circle and a chord. They come back with Dah, yes I know that, and then you build on it. I have done a proof at max 3 times orally before glazed looks are replaced by knowing ones.

I also needed the opposite angles of a cyclic quadrilateral add to 180 which I again mentioned. By now they were trying to get all that they didn't in 9th grade and asked me to prove it. I drew the angles required for proof and told them they could do it themselves.

We found similar triangles in the intersecting chord theorems (it would have been nice if they had asked them to prove it in the book). Here the two triangles APD and BPC are similar and their sides are proportional.

The atmosphere in the class is tremendously different from what we started in the beginning of the sessions. There is a lot of questioning regarding principles. The children are engaged and there is very little idle chatter. When it is there. I pause the class for the children to finish and I have reaffirmed that its not because they are disturbing the class, its because the rest of us don't want them to be left out. The children also listen when another child is talking and don't try to cut them off.

We are done with the 'theory', now to focus on application of the same.

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