Batch 3 - Class 139 - Visualizing Series and more

• MartinShCol 5.5 - From one corner of a square, extend two lines that exactly trisect the square's area - Into what ratios do these lines cut the two sides of the square?

• Answer: 1/3 of each side. If you look at the diagonal originating from trisecting vertex, and look at one half of the square resulting from such a diagonal, the areas within that must be 1:2 ratio. However, the base of both those triangles is the same, so the height must be in 1:2 ratio.

• Solve for 1/2+1/4+1/8+1/16+... - Note that this is an infinite series
• Answer: 1
• We could see this as a linear visualization as under

• Or, we could see it as a 2-D visualization

• Can you now do a finite sum, say we just wanted to do 1/2+1/4+1/8+1/16?
• Answer: This is same as 1-1/16 

• Solve for 1+3+5+7... (2n-1)
• Answer: n^2 (Instructor Notes: If kids are not able to get this straight off, provide a hint by drawing a square grid)

• What would happen if we were trying to find the infinite series sum 1+3+5+7...?
• Answer: Since the square keeps getting larger, the series is divergent (tends to infinity)
• Solve for 2+4+6+8...
• Instructor Notes: Lets kids think about this. If they are not able to make progress, tell them to think about this problem in context of previous problem

• Solve for 1/4+1/16+1/64...
• Answer: 1/3

• (536Dudeney 295) A train was going from Delhi to Mumbai. A wise mathematician claims that while most of the train is going from Delhi to Mumbai, parts of it are all the time going to Delhi! Seriously!? You mean when we cycle from home to school, parts of us are going home? Hey, we didn't say anything about cycling. Can you figure out what's going on?
• Answer: Because there is a part of the train's wheel that is actually below the line of motion (i.e. the rails), those part in fact are always going backwards. The path traced by these is called a Trochoid curve (unlike a cycloid on bicycle, which always moves forward)