Batch 2 - Class 268 - Visualizing Identities and Series

• Hang a hoop on the ceiling at your house. Now throw a ball so that it passes through the hoop. Now hang another hoop behind the first one (at last two feet apart) so that the ball can now pass through both the hoops (you can use this as a game in your school fair)

• Ask kids to visualize simple operations  such as addition, subtraction, multiplication and division

• Visualize (x-4)*(x+2)
• Compute 96*1002
• Visualize a^3 (cube of a) in 3D and 2D
• Ask kids to visualize (a+b)^3 (cube of a+b) - in 3D
• Can kids translate this visualization into 2D?

• 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

• 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.